How Fast Are the TIE Fighters in That Star Wars Teaser?
— Star Wars (@starwars) October 19, 2015
Everyone was excited to see the new Star Wars: The Force Awakens trailer, but I found this short teaser video more interesting. In it, you can see TIE fighters flying overhead. What can we learn from this shot? Can we determine the speed of a TIE fighter? Can we get anything else?
Since this is a video, I will use my standard video analysis techniques. The basic idea is to mark the location of an object in each frame to get position and time data. Since the camera doesn’t pan or zoom during this fly by scene, I just need something to determine scale. Different TIE fighters are at different distances, so I will just use the width of the first TIE fighter as the scale. Assuming these new TIEs are the same size as the classic ones, I will go a width of 6.4 meters.
Now, marking the location of the first TIE fighter in each frame, I get the following plot (where x is in the direction of motion).
Since this is a position-time graph, the slope of the line will tell you the change in position divided by the change in time, which is the velocity. The best fit line gives a TIE fighter speed of 67.3 m/s (150.5 mph). In a previous analysis, I estimated the speed of a TIE fighter at around 200 m/s.
What about the other TIE fighters? Each one is at a different distance from the camera, so I would have to rescale the analysis for each spacecraft. Instead of doing that, let me just look at the angular size (as a percent of the width of the screen) vs. the angular speed. Consider the angular size of an object with a width L a distance r away. I can write the angular size as:
Of course this assumes that the width of the object is much smaller than the distance to the object (which seems to be true for this case). If I take the derivatives with respect to time for both sides of this equation, I get:
Here ω is the apparent angular velocity. I can solve both of these equations for r and I get:
If I assume all these TIE fighters have a similar velocity and length then a plot of ω vs. θ should be linear. Here is that plot.
So it appears that are all traveling around the same speed (except for that one guy).
Now I can do two things. First, I can estimate the change in height of this flying formation. If I knew the actual field of view, I can use the apparent angular size to calculate the distance. I will just assume a field of view of 39.6° like I did with the Avengers 2 movie. From that, the closest TIE fighter is 98 meters away and the farthest is 652 meters. That’s a fairly large spread—although I’m not sure what the official New Order tactical flight formations calls for.
Second, I can find the size of the other spacecraft (the rectangular looking ones). I guess they are some type of troop transport. If they are also traveling at the same speed as the TIE fighters, I can calculate their length. For the two transports, I get 19 meters and 20 meters. I can just average these and say the transports are 19.5 meters long—but I still don’t know exactly what they are.
But why do these spacecraft seem to pass each other if they are all traveling the same (or about the same) speed? The simple answer is parallax. Normally we think of parallax as the apparent change in position for objects of different distances when your viewing position changes. My normal demonstration for parallax is to have someone hold their thumb out at arms length in front of their head. Close one eye and note the location of the thumb with respect to background objects. Next close your eye and open the other one. Notice that the thumb appears to change position. Here is a diagram of your thumb and eyes.
For the TIE fighters, your eyes aren’t changing position but the spacecraft are moving. The effect is essentially the same in that the closer objects appear to move more than the background objects—even though they are all traveling with a similar speed.
Here is another demo you can do. Hold out your thumb in pointer finger in front of your face and move them across your field of view. Your thumb appears to move faster than your finger since it is closer—just like the TIE fighters. I made a gif showing this (since it’s difficult to explain).
One last thing. If you really understand something, you should be able to make a model of it. So, that’s just what I did. Here is a python model (using trinket.io) showing some boxes (representing the TIE fighters) moving at a constant speed but at different distances from the camera.
If you like that, take a look at the code. Make your own version. It’s not too difficult.
OK, let’s summarize this analysis.
- These TIE fighters seem to be going about the same speed and around 67 m/s.
- The flight formation appears to be flying from 90 to 650 meters above the ground.
- If those other objects are flying with the same speed as the TIE fighters, they would be between 19-20 meters long.
- Yes, it seems as though I can make a model that shows objects traveling at similar speeds but different distances from the camera.
- Why am I doing such a silly analysis? The only reason is that I think it’s fun. The end.