The Lord of the Rings trilogy came on TV again recently. My wife and I can’t help but to watch this even though we have it on DVD. Anyway, I was thinking about the part where Gondor sends a signal to Rohan asking them for military aid. Since this was before the invention of email, they had to do it with a signal fire. Hopefully this clip won’t be a spoiler for you, but this is from the movie. Actually, all of the clips available have embedding disabled. So this is just a picture of that clip. If you want to see it, go to the youtube version.
One other spoiler: Sauron is really Frodo’s father. Ooops. I really shouldn’t have let that one out.
How fast does this signal travel? There are three angles to this question. Symbolically, what is this speed? What is my estimate of the speed from the video clip? What would be the speed if someone really set this up – you never know, you might need to do this in the event of a zombie attack.
I was going to call this “theoretical” speed, but that might be confusing. Instead, I am just going to set up the problem without putting any numbers in. Let me start with a picture. Also, let me assume that the signal fires are all equally spaced.
The distance between signal fires doesn’t matter so much as how many signal stations are there. If I use the variables form above, then the number of stations will be:
What about the time? There are two times. First, how long does it take for people at a station to react to the previous signal fire? I will call this tr (reaction time). The other important time – how long does it take to light the signal fire? I will call this tl. To calculate the speed of the signal, I need the total distance and the total time. This gives:
Now, substituting for the total time and the total distance, I get (in terms of n):
This solution tells me what I need to know. The average distance, the time to light the fire and the reaction time. Note, there is already an assumption. The first assumption is that there are enough signal waypoints that the ends don’t matter. Actually, I guess even if there is only one fire, it still takes time to light and for the receiver to react. Ok, then that is not an assumption.
From the movie
Can I get an estimate of these three parameters from that movie clip? Oh, yes I can. Will it be realistic? Who knows. Here is the stuff I found.
- When Pippin lights the first signal, it takes about 12 seconds from the time he puts the fire on it until it is mostly lit.
- After the first signal is on fire, Gandalf sees the next signal only 6 seconds later. WHAT? The guys (or gals) at the next station must have just been sitting there staring and waiting for a signal. Oh, it was probably like 40 years since the last time it was used. I guess you can do stuff like that if you don’t have youtube. But wait, the more I think about this, the more upset I get. I am ok with invisible rings, flying dragons, glowing swords and stuff. However, it is beyond the bounds of reason to expect me to believe that some guys are sitting way on the other mountain with a hair-triggered lighting mechanism. Six seconds. Seriously.
- The next time to light is 12 seconds. That is reaction plus light time.
- The next one is at night and has a total time of about 6 seconds. At night! Don’t these guys even sleep?
- 3 seconds for the next one. Come on man.
In this last one, Aragon notices the signal in under 2 seconds. Luck or skill?
A couple of other things I noticed. The first fire had a roof over it – very sensible. The other fires seemed to be open to the air (and rain).
Also, it seems like all of the fires are on tops of mountains. At least they didn’t show any on flat ground. One other thing, the signal started during daytime, went through the night and ended at day time. I am not sure how far apart these two locations are (Rohan and Gondor), but I doubt this signal went faster than the rotation of Middle Earth. How fast does Middle Earth rotate anyway? Well, it seems there are a couple of possibilities here.
- The signal speed is fast, but the distance is very far. So far that it takes a long time to get there (more than 1 night).
- The signal speed is uber fast and Rohan is on the other Side of Middle Earth. The signal passes through the shadow side of Middle Earth, but takes less than one night to get there.
- The night is really short on Middle Earth
Didn’t Gandalf and Pippin ride from Rohan to Gondor without stopping? It couldn’t be THAT far even on a super-horse. If the horse went about an average speed of 15 mph for 30 hours straight (just my first guess) that would be 450 miles.
Back to my estimations. So far, from the movie I have that the total time for one signal seems to be on the order of 10 seconds. What about the distance? My first guess is on the order of 50 miles. I just guessed that. Complete guess. Well, in the movie, all the signal fires are on mountains. There aren’t too many mountains in this part of Louisiana (there aren’t really any rocks even). So, here is a google map of some peaks near Vail, Co (the first thing I could think of).
From this map, the nearby peaks are only about a couple of miles away. Ok, now I am looking at Pikes Peak. Still, the nearby stuff seems on the order of 5 miles away. Ok, I am going with 5 miles.
Putting this in, I have a signal speed of:
Correction Note: I fixed the above expression. Originally, I had this as 18,000 mph. It should be 1,800 mph. The mistake was pointed out by commenter “some guy”. Thanks “some guy”. Also, this would change some of the statements below.
Wow. If I use that speed and assume that it takes 12 hours (because it goes through the night) then Rohan would be
216,000 21,600 miles away. Note: the circumference of Earth (not Middle Earth) is around 24,000 miles.
Ok, maybe that part of the movie is wrong. What if it is 500 miles away? How long would the signal take? There would be about 100 signal stations (at 5 miles apart) and each one would have around a 10 second turn around time. This would be 1,000 seconds or around 15 minutes.
A more realistic estimate
What if I forget about the movie? How fast could I send a signal from say New York to Raleigh, NC – about 450 miles? Clearly, I need to estimate some stuff.
I think this is the toughest to estimate. How far away can I put a fire and still have someone see it? Obviously, terrain matters. In flat parts of the Earth, you might be hard pressed to see 5 miles. But, I think I could make a fire on a mountain top that could be seen perhaps 20 miles away. Really, the question is: how far away could you see a fire? To test this, I took some pictures of a candle flame. Yes, I did. Here is the flame (about 1 cm tall) from about 4 meters away.
Here it is about 15 meters away (I zoomed in the image after I took the picture – so I did not use any optical zoom.)
And now at 30 meters.
I can’t really see this at 30 meters in the picture – but with the naked eye, I could kind of pick it out. So, that is one candle at 30 meters away. Let me assume that has a light power output of Po. This would give an intensity of light at that distance of:
I am calling this Id. The d stands for detect – thus the lowest intensity that I can detect. Now suppose I ramp this flame up to a big bonfire. If it is about 2 meters tall (and wide), and if I assume the power output it proportional to the surface area, then this bonfire would have an area of about:
I assume if it has 40,000 times the area, it will have 40,000 times the power output of the candle. So, how far away could someone be and have the same intensity (assuming the light is even radiated in all directions).
So, that seems that I could see this about 19 km or about 11 miles away. That seems pretty far, but ….. maybe I should use something a little closer for the average distance. 13 km seems like a good average.
Note: I know you can see light pretty far away. For example, take a light house. If you are out at sea, these suckers can be seen at least 10 miles out (if the tower is tall enough). Also, the Sun. It is really far away, but I can see that (Double Note: don’t look at the Sun, you could hurt your eyes.)
What about the reaction time. I guess this could be really short, but I am imagining a station with 2 people in it. If you only had one person, the reaction time could be much longer. With 2 people, they could take turns sleeping and they could help each other do chores (also, they would have someone to play checkers with). Reasonably, I think a reaction time of 10 minutes on average seems like a good guess. If I were sitting around for years waiting for a signal, I would probably just keep checking it every once in a while and not just stare at it. Oh, and what if there were clouds or rain? It could feasibly be the case where someone doesn’t notice the fire for around an hour.
Lighting time. Technically, this could be quite short (or the order of a minute). However, things could happen to make this much larger. Suppose it rained or some lighting error occurred. Maybe the chemical that was put on the fire “wore out” or something. It could take 30 minutes or more to get this sucker going. With this in mind, I am going to use a lighting time of 5 minutes on average.
This will give a signal speed of:
So, back to the New York to Raleigh thing. How long would this take? To send a signal 450 miles, it would take 14 hours. I like that answer. But, here is the problem. How many signal stations would you need? Over 50 stations. That is 50 stations that need to have people and supplies. Maybe in the grand scheme of things, that isn’t too bad.
Final Note: What about the book? I found my Return of the King book and looked it up. It seems that Pippin doesn’t like the Beacon of Gondor. Instead, he notices it as he and Gandalf ride to Gondor. The only details I found was that they could see the lights traveling as they rode.