In keeping with my long tradition of doing some kind of physics analysis for Star Wars Day (May the 4th Be With You), I celebrate this year by looking at the Death Star II in Return of the Jedi.
In a pivotal scene, the Rebels mount an all-out offensive against an enormous Super Star Destroyer in the battle over Endor. The attack on the bridge sends the ship crashing into the Death Star. That's about all you need to know.
I don't know why the Destroyer crashed after a fighter hit the bridge. Surely a vessel so vast would have a backup bridge. But knowing why is not my job. Instead, I'll focus on look at the time between when the crew lost control of the Destroyer until the time it hit the Death Star.
Let me start with two important details. First, the size of the Super Star Destroyer. I won't try to estimate this, I'll just use values from other experts. Wookipedia pegs the Executor-class Star Destroyer at 19 kilometers long. The second important detail: Death Star has a diameter of 160 kilometers.
Next, there are two things I need to determine from the movie. First, the altitude (over the Death Star) of of the Star Destroyer. Honestly, this is rather difficult. We never get a clear shot of both the spacecraft and the Death Star. Really, the best view shows the Death Star as seen from the bridge of the Star Destroyer just before it crashes.
It is entirely feasible to use the apparent curvature of the surface of the Death Star to estimate the altitude of the starship. However, it still would be an estimate, and it would include some fairly complicated calculations (granted, there is a possibility that it isn't too hard and it's my fault). Instead let me just say that the Death Star looks similar to Earth as seen from the orbital altitude of the International Space Station. I am going to assume that the ratio of altitude to "planet" radius is the same for both cases. That means:
Using an ISS altitude of 300 km and and Earth radius of 6,370 km, I get a Star Destroyer altitude of 3.77 km. Of course that can't be correct since the length of the Executioner-class Star Destroyer is 19 km. OK, let's just hold off on this value for now.
Now let me focus on the impact speed. If I use the length of the Star Destroyer, I can get the position of the spacecraft in each frame using Tracker Video Analysis. Looking at just the motion toward the Death Star, I get the following plot.
The slope of this plot shows an impact speed of 3.5 km/s (7,829 mph). Yes, that's pretty fast.
The real question remains---why is it moving so fast? There are three possible answers:
- After rebels destroyed the bridge, the Super Star Destroyer veered out of control and used its thrusters to drive into the Death Star.
- The Destroyer used its engines in some way to stay above the Death Star. The attack eliminated this ability, and the ship fell into the Death Star due to the gravitational interaction between the two objects.
- The impact was the result of the engines and gravity.
For the purpose of this analysis, I am going to assume the collision was due only to the gravitational interaction. If that's the case, I can use this to estimate the mass of the Death Star. I will make a couple more assumptions:
- The Super Star Destroyer starts this motion from rest (velocity = 0 km/s).
- The starting altitude of the SSD is 30 km above the Death Star.
Since we don't really care about time during this motion, we will use the Work-Energy Principle. This states that the total work done on a system is equal to its change in energy. If I include both the Destroyer and the Death Star in the system, there will be no work and the energy will include both kinetic energy and potential energy. I can write this as:
One important note. Technically, both the Death Star and the Star Destroyer will have changes in kinetic energy. However, if we assume the mass of the Death Star is much greater than the spacecraft it will have a negligible change in velocity. From the work-energy equation, I can solve for the mass of the Death Star (I am calling that mass-2).
G is the gravitational constant (6.67 x 10-11 N*m2/kg2) and v2 is the velocity of the Star Destroyer on impact. Using my value for the impact velocity and the starting and ending position (with respect to the center of the Death Star), I get a mass of 2.7 x 1022 kg. That would give it an average density of 1.25 x 107 kg/m3. If the Death Star was solid steel, it would have a density around 8,000 kg/m3.
Yes, this causes some problems. First, how do you get the density that high? Maybe the Death Star has a super dense inner core---maybe. Also, this large mass along with the relatively small size would make the gravitational field on the surface very large, 28.7 times larger than the Earth's surface field. Second, the above plot of position shows a nearly constant velocity. With this large of a mass, the Star Destroyer would have a non-constant acceleration as it moved closer to the Death Star.
Yes, I know there is another explanation for the large calculated mass of the Death Star. The other reason is that Star Wars is just a movie and the Star Destroyer crashes because it is a model controlled by humans. Honestly, I am OK with this explanation---because it's obviously true.
Since there have been stories about how scientists like to the suck the fun out of everything, let me make a few points.
- I'm a huge Star Wars fan. I saw Episode IV in the theaters when I was young and I loved it. I'm also a big fan of science fiction, fantasy and super heroes (not sure if the Marvel Universe would be considered science fiction).
- There are physics errors in just about every movie---and really, I'm OK with that. I don't interrupt the movie to point out these errors, I just watch the movie and enjoy it.
- Yes, it's nice when a movie includes correct scientific ideas but that's not the goal. The goal is to make an entertaining and compelling story (even if it sometimes seems like the only goal is making money). Saying that filmmakers should include better science is like saying scientific papers should have a better plot. Actually, I would love to see scientific papers with an interesting plot.
- Then why do I analyze the physics of Star Wars (and other things)? I like to use scenes from cool movies to explain physics. And sometimes I see something cool like the collision with the Super Star Destroyer and think, "I wonder..." That's it. It's just fun physics.
- But what about a movie that wants to use better physics? Help is always available. The Science and Entertainment Exchange is a great place to start. It connects movie producers with real scientists.
- I have to admit it---sometimes I do get a little bit of geek rage over some science in a movie.
Now for something else.
Oh, you thought this would all be over by now? Wrong. The physics never ends. Check out the Super Star Destroyer right after it loses control.
Let's first look at the rotation of this crashing starship. If I mark the bridge as the point of rotation, I can get the following plot of angular position vs. time.
From the slope of this plot I get a nearly constant angular velocity of 0.159 radians/second. Big deal, right? Yes, it is. Since the Star Destroyer is rotating, everyone aboard is moving in a circular path. To move in a circle, you must accelerate. This centripetal acceleration depends on the angular velocity (ω) and the radius of the circle. The magnitude of this acceleration can be written as:
The angular velocity for this Super Star Destroyer isn't super large---however, the circular radius for the Imperial crew at the front of the vessel will have a super large radius. If I approximate this radius at about 15 km, I can calculate the centripetal acceleration with a value of 379 m/s2 or 39 G's. That might not be a high enough acceleration to kill you outright, but the crew probably passed out before the collision with the Death Star. I guess that is for the best. Who would want to witness that crash and the explosion of the Death Star?
Now you can see the importance of a nice physics analysis of Star Wars. It's not just to point out errors, but to provide more meaning to the plot. Next time you watch Return of the Jedi, just think of all the helpless Imperials stuck in the doomed spacecraft.
