And we’re back. Yes, it’s another impact movie, but for good reason.* ‘Sides, after Outland how can you say no to another bad sci-fi movie starring Sean Connery? This one is Meteor (1979), not to be confused with the 2009 miniseries of the same name (stay tuned for that one).
Now, this film was supposedly based on a real thing, Project Icarus, in which an MIT professor gave his students a challenging assignment: an asteroid that crosses the inner solar system (name 1566 Icarus because of its orbit’s close path to the Sun) has been deflected due to an encounter with a non-periodic comet and is now headed for Earth. You have 18 months to stop it. GO! Their solution? Nuke it! Yes, we have a pack of MIT students in the ’60s to blame for the never ending string of impact movies.
In the film they change the asteroid to Orpheus, which was fictitious at the time. In yet another instance of life imitating art, a new asteroid was discovered in 1982 and christened 3361 Orpheus. The real Orpheus isn’t a main-belt asteroid like the movie Orpheus, but instead crosses the orbits of Venus, Earth, and Mars. In fact, it passed within 40 times the Moon’s distance to the Earth in December 2013!
Much like the real project Icarus, a comet is headed for the asteroid belt on a collision course for Orpheus. To up the drama, a crew of astronauts on their way to Mars happens to be in the area of the asteroid belt and is sent to check it out. Ok, let’s stop and have a look at the asteroid belt, courtesy of NASA. You’ll see this image again in another post soon:
The unit of distance in this diagram is the astronomical unit (AU), where 1 = the distance that the Earth is from the Sun (on average). That’s about 93 Million miles, or 150 Million km. Mars sits at ~1.5 AU. The asteroid belt is ~3 AU from the Sun, or another 150 Million miles from Mars, assuming Mars and Orpheus are even on the same side of the Sun at the time! That’s a heckuva detour!
Not to get too sidetracked here, but I just read the excellent book “The Martian” by Andy Weir. Mr. Weir does an excellent job of, if not thoroughly explaining, at least hinting at the difficulties of orbital mechanics. Space is big and, even in a fictitious world of inexhaustible fuel supplies, moving from one orbit to another is tricky. Let’s say you’re orbiting Mars. Mars, in turn, is orbiting the Sun. So is Orpheus. Mars is moving, Orpheus is moving, and you’re moving, so going from one orbit to another is like trying to hit a moving target from a moving platform. You have to correctly predict where the target is going to be at the time of your arrival. If you miss, you might get lost in space forever. In the film, the comet is discovered when it is about 400,000 km from the asteroid belt. A typical periodic comet will cover that distance in about a day when it’s 3 AU from the Sun. For the Martian astronauts to get there in time to witness the comet hit Orpheus, they’d need to travel at just under 1% the speed of light, assuming instant acceleration/deceleration and minimum straight line path. But the movie tells us they need just under 3 days to get there in time to see the impact, so that means we now know something about the comet’s velocity–it’s moving at around 5500 km/h. Which means our astronauts only need to travel at around 3 Million km/h to get there in time, and since that’s only 0.27% the speed of light at least we don’t have to involve relativity in the calculation. You know how you could avoid this whole problem, screenwriters? Don’t make them Martian astronauts. Say they were on a mission to explore the asteroid belt. Why does Mars even enter into the conversation? (My guess? Laziness.)
After the comet dislodges the asteroid and kills the astronauts, it’s headed straight for Earth. We’re told this story in flashback from a few days ago and also told that Orpheus will hit the Earth in 6 days, so let’s say 10 days to travel 2 AU, or a bit over 1% the speed of light. This is right around the velocity where you start to have to include relativity in the calculation (thanks a lot, Einstein!) but we’re going to vigorously ignore that for now.
Time for some math! Don’t worry–it won’t be too bad. A typical comet has a mass of a few times 10^13 kg but we’re told that this one is big, so let’s say it’s 5 x 10^14 kg to be generous. (That’s around 2 times the mass of comet Halley, for comparison.) There are a few assumptions that need to be made in order to know how much force is imparted to the asteroid. Here are the very generous (read: bad) assumptions I’m making. 1. The collision is completely elastic, so 100% of the energy is transmitted from the comet to the asteroid. 2. The collision is instantaneous, so the force is just magically transferred from the comet to the asteroid. 3. No energy is lost when the asteroid hits the space ship. 4. Here’s a big one: the mass of Orpheus is 5 x 10^18 kg. We’re told that Orpheus is big, but nothing more, so I’m assuming it’s around 100 km across and thus one of the 100 or so largest main belt asteroids.
This is an extreme oversimplification of the problem, but represents the best-case-scenario for the movie so we’re going to go with it. We’re assuming 100% of the kinetic energy of the comet is transferred to Orpheus. We can do two different calculations here: 1. How fast is Orpheus traveling given the known velocity of the comet determined from the movie? 2. Given how fast Orpheus need to be traveling to reach Earth in 10 days, what velocity would the comet need to have had when it hit Orpheus. Both of these are just problems of setting the kinetic energy of the comet before impact equal to that of Orpheus after impact. Kinetic energy = 1/2 mv^2.
1. Plugging in the numbers for the first scenario tells us that, after the impact, Orpheus is traveling at 55 km/h toward Earth. That gives us just over 620 years to get ready, which is pretty reasonable, I think.
2. For the second case scenario, the picture is much more grim. Since we need to get Orpheus up to 1% the speed of light, our comet needs to hit the asteroid traveling at… just over the speed of light. Crud. We do need to include special relativity in the calculation. Well I gotta put my foot down somewhere and it’s not into a steaming pile of special relativity. We’re going to have to say this one’s impossible. So which variables can we play with to make it more possible? We can’t adjust the mass of the comet–we’re already about as massive as comets can get. We can, however, play with the mass of Orpheus.
Later in the film, we’re told that the biggest piece of Orpheus is 5 miles wide, rather than the 100 km I had assumed. Asteroids in that size range have only poor mass estimates, but they’re thought to be between 10^15 and 10^16 kg. If we assume that 100% of the energy of the comet is transferred to just this one 5-mile chunk of Orpheus, and if we assume that chunk is 5 x 10^15 kg, then our comet only needs to be traveling at a pokey 11,000 km/s, or a bit more than 25% the speed of light. We still have to invite Einstein to the party, but at least he can relax and have a glass of punch.
As with many of these movies, the science is mostly front-loaded, and most of the film is just implementation of the stupid established plot. So there’s not much to comment on from here. They’re going to nuke the rock thanks to a space-based missile platform, originally designed for precisely this purpose by Sean Connery, but since taken over by the military to set their sites on more terrestrial targets.
Why are these missiles so long? Rockets look like this because they’re mostly fuel used to break free of Earth’s atmosphere. Going down is the easy part–Newton provides the acceleration for free!
We’ve gone through the exercise before of ridiculous nuke sizes, mass-to-yield ratio, the size of the largest earthquake ever, etc., so I don’t need to tell you that we’re doomed. Only combining our space weapons with those of the Soviet Union do we have any hope of survival. The original plan of Project Icarus was to just give the asteroid a nudge so that it misses Earth. So do we make it, or is this the end? Spoilers…we make it. They completely pulverize Orpheus which, since it was traveling at 25% the speed of light, caused high energy particle showers that vaporize Earth’s atmosph… no wait, that was my dumb calculation. Never mind.