Dot Physics

How fast is the Beacon of Gondor?

Posted by Rhett Allain on July 30, 2010
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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.

i-0ffcdb022a0a429812af258e5da7c2ff-2010-07-28_clip_1_1.jpg

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.

Symbolically

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.

i-4503054ae92bb9e53bc08121a9266089-2010-07-28_untitled.jpg

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:

i-af4d8f57e1113b7a3981b4887c4d431c-2010-07-29_la_te_xi_t_1.jpg

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:

i-fedbde0d9374b28066f102941d8439aa-2010-07-29_la_te_xi_t_1_1.jpg

Now, substituting for the total time and the total distance, I get (in terms of n):

i-1776a6c5956646ff7f34c7413f4541a4-2010-07-29_la_te_xi_t_1_2.jpg

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?

i-a477fb76ed28ea5f76b08441a1001013-2010-07-29_tracker.jpg

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).

i-f777de3bead32dc29d15fe9da15d36ee-2010-07-29_untitled.jpg

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).


View Larger Map

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:

i-5f4c1398741bce4a3ed86bfbc059f2a7-2010-08-08_la_te_xi_t_1_23.jpg

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.

Distance

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.

i-1f4342cf1674f0d7a4e4dadbf63f296c-2010-07-29_flame_1.jpg

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.)

i-9f5f725e55587563ae37ddcb503a7897-2010-07-29_flame_2_1.jpg

And now at 30 meters.

i-32f77e5f1e763a7e8d650d1d4fbda4a2-2010-07-29_flame_3.jpg

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-bffc40ba32e373ae916cca7773e8a84e-2010-07-29_la_te_xi_t_1_5.jpg

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-5364019458f3b0904653366535a46f57-2010-07-29_area_ratio.jpg

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).

i-c3bdbad7d549fdaa0cf4c06487fd2c57-2010-07-29_bonfire_distance.jpg

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.)

Time

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:

i-1e0ce3a5a0290b6fec5b3dc821a81f3b-2010-07-30_la_te_xi_t_1_7.jpg
i-825a11911564ea6b4b97c690d8ea136c-2010-07-29_la_te_xi_t_1_6.jpg

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.

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Comments

  1. #1 Fran
    July 30, 2010

    My question is, how long did it take you to put together this blog post: writing, getting the stills from the movie, adding in the equations, lighting candles and taking photos?
    It looks like you are enjoying your summer!

  2. #2 Eric Lund
    July 30, 2010

    This system sounds similar to a Napoleonic semaphore line. The Wiki article mentions that average transmission time between Paris and Lille (230 km over mostly flat terrain) via 15 stations was 32 minutes (weather-dependent, obviously), and that was for a message with content. Of course this was with trained military personnel specifically looking for transmissions from the neighboring stations on the line, and did not involve lighting fires. At its peak, the French system included 556 stations covering about 4800 km of lines.

    Comparing with your New York-Raleigh estimate, the required number of stations is close (in the high 40s), but typical transmission times would be much shorter, around 90-120 minutes. These semaphore systems also have the disadvantage of being pretty much unusable at night, unless the system included a fire/acetylene torch component.

  3. #3 djlactin
    July 30, 2010

    It’s MOOOVIE! Suspend your incredulity for a few hours. (Also, the movie departs seriously from the book on many occasions.) The book is far more entertaining. (Read it 3 times so far.)

  4. #4 Mike Dunford
    July 30, 2010

    I don’t have any problem with the length of time it took for the signal to be noticed at each station. Watching for the signal is their job, and presumably everyone knows it’s a time of high tension, and there’s a reasonable expectation that the fire might be lit soon.

    For comparison – I supervise lifeguards. Lifeguards are expected to recognize a drowning swimmer within 10 seconds. We test this by having swimmers release a rubber duck and record how long it takes the guard to notice the duck (they signal when they see it). That requires them to notice the appearance of a new object in what is frequently a crowded field of view, and our guards manage this in 10 seconds more frequently than not. The fire appearing should be an easier spot, since it’s going to involve major, eye-attracting motion.

  5. #5 AnyEdge
    July 30, 2010

    You have forgotton one clear and crucial element: for narrative structure, the period between lighting one beacon and the next may have been shrunk by the filmmakers. So, while it may appear that only 12 seconds pass between the lighting of one beacon and the lighting of the next, in reality, it might be reasonable to calculate, say, several hours. The only one that mattered was the first to the second (because this was the unauthorised lighting, it had to be noticed before being extinguished.). After than, hours might elapse without dire consequences.

  6. #6 Martin Madsen
    July 30, 2010

    Great post! As a physicist and LOTR fan, I love this kind of in-depth analysis. Given that Tolkien was not much of a scientist, I’ve found he got stuff wrong in a number of places. However, I think your real-world estimates here are reasonable- 14 hours or so to send a signal about 500 miles is close to Tolkien’s story line. The maps have Minas Tirith and Edoras about 100 leagues (345 miles) apart. If I remember correctly it took Gandalf about 3 days to ride that. If they can get a vital signal across that distance in 1/2 a day, that is a significant improvement in communication time.

  7. #7 Ben
    July 30, 2010

    You’re also assuming that all the other lightings were close to the average speed. Perhaps it shows it at night because at one or more stations the watchers were asleep, or had gone out for the afternoon. Maybe they were fighting orcs or reciting some really tedious elvish poetry. So you have this extremely well-transmitted message with a couple of six-hour speedbumps in it.

  8. #8 Mu
    July 30, 2010

    Well, they say it’s six days of hard riding from Rohan to Gondor. A good horse can do 100 miles in 24 h, this being a movie, it can do it 6 times in a row. This makes the maximum distance 600 miles. Since the signal traveled through the night at very high speed, it must have made it around 1/4 of the planet (at high latitude you might have a short night, but not so high as not to have a real night). So the circumference of middle earth must be significantly smaller than earth. Since the gravity seems close to standard, you’ve just proven middle earth has a lead core.

  9. #9 AnyEdge
    July 30, 2010

    @8: Shadowfax is far, far more than a mere ‘good horse’.

  10. #10 Aaron
    July 30, 2010

    Just to be geeky. The distance from Edoras (Capitol of Rohan) to Minas Tirith (Capitol of Gondor) is about 270 miles. And they are pretty much at opposite ends of a mountain range (The White Mountains), so there is a nice line of peaks between them.
    Also, Middle Earth is supposed to be Earth, so when wondering about the length of day and night, you can use real world answers.
    Also when Gandalf and Pippin ride to Gondor, they ride on Shadowfax, a horse that does not tire (it’s magic, dontchaknow) so a horse galloping at about 35 mph over various terrain gets there in 7-8 hours.

  11. #11 Who Cares
    July 30, 2010

    There is a real world example you can compare this with.
    The great wall of China had this warning system and could pass a message across it’s length within a day.
    Big difference was that the people on the wall there were motivated to keep a lookout (the other choice being dead) compared to people who’ve been waiting for 40+ years to light a fire that for all they know they might never need to light.

  12. #12 the Rising Jurist
    July 30, 2010

    Some book-nerd notes:

    The established distance between Edoras and Minas Tirith is closer to 275 miles, as the crow flies (350 miles by horse, via the Great West Road).

    Also, the book lists only seven beacons, excluding the one in Minas Tirith itself. I guess fires are hyper-visible in the White Mountains air.

  13. #13 CalcDave
    July 30, 2010

    I think the Great Wall of China was meant to have signal fires lit along it to pass similar information across. I bet more real-world data could be gathered from that realistic situation.

  14. #14 Tim H
    July 30, 2010

    According to LotR, the northern beacon system consisted of 7 stations extending west from Minas Tirith: (east to west) Amon Din, Eilenach, Nardol, Erelas, Min-Rimmon, Calenhad, and Halifirien. (There was also a southern system to warn the seaward provinces.) Using the map from Unfinished Tales, which has a scale and is approx 50% larger than the paperback LotR, the path the beacons cover would be around 160 miles long. Of course the system ended at Halifirien on the Gondor-Rohan border, so unless the Rohirrim relief force is camped there waiting for a signal to cross the border your signal must be carried by mounted rider to Edoras, which is about another 150 miles. Minas Tirith to the Fords of Isen is about 150 leagues. Osgiliath to Bree by the Numenorean road is stated as being 392 leagues in the essay on the Disaster of the Gladden Fields.

  15. #15 Rhett Allain
    July 30, 2010

    @Fran,

    Yes – it took some time. However, I am not really enjoying my summer because it has been too busy with not enough time for blogging. But, summer classes just ended, so maybe I can get some stuff done.

  16. #16 bsci
    July 30, 2010

    Knowing Tolkein’s background, I suspect he modeled the fires after the 4th century B.C.E. Jewish tradition of lighting fires from mountaintop to mountaintop to signal the new month (Rosh Chodesh). The fires were lit when a silver of new moon was visible in Jerusalem and the final fire was visible in Babylon (Iraq). The distance was about 500 miles.

    I couldn’t find enough information at the University of Google to make a useful calculation, but it was well less than a day (until protesting groups starting setting other beacon fires to try to confuse the system)

  17. #17 Rhett Allain
    July 30, 2010

    To all:

    Let me just say that these are great comments. This is why I write this kind of stuff. Nothing like a good geeky conversation about physics and Lord of the Rings.

  18. #18 Kitty
    July 30, 2010

    Should have used the mountains in New Zealand for reference, as they’re where the movie was filmed.

  19. #19 Tim H
    July 30, 2010

    In the passage where Pippin sees the signal fires in Return of the King, Gandalf states that each beacon location is also a relay station for errand riders, capable of providing fresh mounts. Given that the northern system only extends halfway to Edoras, that Rohan needs more info and deserves more diplomatic courtesy than a contentless signal fire could provide, and that the formal request for aid was is fact delivered by errand riders, it is likely that the beacon system was only used to warn Gondorian residents of Anorien (the province along the first 150 miles of road, up to the border with Rohan). It seems unlikely that Anorien would be uninhabited in peacetime; warning the population would allow them to head for refuges in the mountains and send reinforcements to Minas Tirith.

  20. #20 Tim H
    July 30, 2010

    As for the speed of Shadowfax, he covered about 450 miles (Dol Baran, just west of the Fords of Isen to Minas Tirith) in about 78 hours (0000 March 6 to about 0600 March 9). He stopped for a few hours at Edoras (call it 6). 450miles/72hrs = 6.25mi/hr, between a walk and a trot. Shadowfax is described as galloping hard. Tolkien’s math is off on this one.

  21. #21 Steinn Sigurdsson
    July 30, 2010

    The beacon system is based on the English beacon system, on the Malverns, which goes back at least a 1000 years and was famously used correctly in the 16th century when the Spanish Armada approached.

    The movie series of beacon light-ups should maybe be taken as artistic license, compressing the sequence in the interest of audience patience. Otherwise there’d be an entire fourth movie consisting of them sitting there watching to see how quickly the next beacon lit.

  22. #22 MartinDH
    July 30, 2010

    bsci:
    Rather than Jewish tradition, I suspect J.R.R. was more influenced by the old beacon system of Southern England. These beacons were used from Anglo-Saxon times (9th c…warning of Viking landings) to the end of the Napoleonic wars.

  23. #23 Shawn Cornally
    July 30, 2010

    This is why the Internet is for more than world of warcraft. You just solved my years-long quest for finding some awesome way of teaching radiation intensity. Thank you, sir!

    =shawn

  24. #24 NoAstronomer
    July 30, 2010

    @Steinn

    I grew up in the shadow of the Malverns and I hiked along the ridge countless times. For the 400th anniversary, in 1988, of the Spanish Armada the beacon line was recreated through the Worcestershire Beacon (highest point of the hills) to points beyond.

    I recall watching the TV coverage and seeing the beacon lit from my home. I don’t remember where the first beacon was, probably in Cornwall or Devon, but it took around 50 minutes before I could see the lit beacon about 5 miles away.

  25. #25 Shar
    July 30, 2010

    This is very interesting…sort of. But I have a question for you…since when is anything in a movie in real time? I mean, they had to cram a very large and detailed story into a 3 or so hour segment. The battles for instance, they went on for days…do you want each movie to be that long as well? I’m in love with Tolken’s work and I think they did a pretty excellent job with the movies (other than the hobbit’s feet and homes, in my humble opinion) but there is no way to make the movie as real as you’d like it to be. Just throwin the obvious out there…

  26. #26 Bing
    July 30, 2010

    You, sir, are one awesome nerd!

    HJ

  27. #27 Daniel J. Andrews
    July 30, 2010

    Love this blog post! I thought the beacon system was unreliable not because of the timing (good points though) but because I was thinking of the beacon environment.

    I’ve worked a fair amount in mountains and have camped/worked near the top of many on the west coast. We’ve gone days without seeing the sun, been stranded more times than not when the helicopter couldn’t pick us up due to bad weather. Plus, snowcapped mountains can get a lot of snow and high winds…sometimes it would be an hourly job shoveling snow drifts away from those beacons (assuming they aren’t whipped clean by high winds which make it suicidal to try and walk around in).

  28. #28 feralboy12
    July 30, 2010

    There seems to be some disagreement over the length of a “league,” the common unit of measurement in Middle Earth. My research, which consisted of reading “Bored of the Rings,” shows that a league is approximately three furlongs, or about a knot short of a hectare.

  29. #29 Tim H
    July 30, 2010

    The English “league” generally means 3 miles. It is supposed to be equal to the distance an average person could walk in 1 hour. In Middle Earth, “league” is used as the Westron translation of the Numenorean “lar”, which is basically the same thing. There are 5000 rangar in one lar. A rangar is the length of an average Numenorean stride from rear heel to front toe, approx 38 inches, which makes a lar 5277 yards. (Numenoreans where taller than modern man. In later years, average height, Man-High, was two rangar, 6’4″, since they had mixed with lesser men. Galadriel was man-high. Elendil the Tall was said to exceed man-high by 1/2 ranga, so would have been 7’11″. That’s why they called him Tall.)

    from Unfinished Tales–Disaster of the Gladden Fields- appendix: Numenorean Linear Measures

  30. #30 Jason
    July 30, 2010

    Rhett, you say “One other thing, the signal started during daytime, went through the night and ended at day time.”

    Isn’t it possible that there was a solar eclipse (which you have mistaken for night)? A lot of things fall into place if this were the case.

  31. #31 riri
    July 30, 2010

    You have unbalanced parenthesis in one of your equations. For every unpaired bracket, god kills a kitten…

  32. #32 Rhett Allain
    July 30, 2010

    @Jason,

    That is brilliant. I didn’t think about a solar eclipse.

  33. #33 Rhett Allain
    July 30, 2010

    @riri,

    If I fix it, will it bring the kitten back? If I don’t fix it, will there be another dead kitten? How long do I have?

  34. #34 Cherish
    July 31, 2010

    I would say that 11 miles would be a good estimate for a pretty good sized fire. I remember that when you’re at White Sands, you can see the Dunn Solar telescope (which has an above ground component that is about 50 feet tall) at the top of the mountains, but it’s just a white speck. White sands is around 20 miles away.

    I also remember being able to see the top of a crane at the south side of Fargo (about six miles away on some of the world’s flattest terrain) that was on the order of 8 stories high…but would have needed a pretty good size fire to see that.

    Based on my observations, I think the spacing would be highly dependent on the size of the fire.

  35. #35 bryce lanham
    July 31, 2010

    you are douche bag….stop ruining the movie

  36. #36 steve
    August 1, 2010

    Did this for real earlier on this year (“Illuminating Hadrians Wall”): flares and/or braziers spaced about every quarter mile from Newcastle to Carlisle, first brazier lit in Newcastle at about 6pm, flame arrived in Carlisle at 6.40pm (the distance was about 150 miles)

  37. #37 Rafael
    August 1, 2010

    A complete aside, not being a LOTR fan (heresy!, I hear you cry):
    You use “sec” in one of your equations… I normally tell my students off for doing that, are physicists around the room Ok with that?
    (Aside on the aside: When I said that to one of my classes, a girl said out loud “Oh, but I always do secs” – at which point the class burst out laughing, and the girl turned a bright shade of crimson)

  38. #38 stripey_cat
    August 1, 2010

    Tim @20 – Tolkien was experienced with cavalry in bodies, not with individual riders, so he overestimates the time taken. (Ditto his speeds for moving men around – fine for armies, but not so good for a fell-runner.) That said, horses need to eat, drink and sleep for a significant chunk of each day if they’re to be fit at journey’s end, which we’re explicitly told Shadowfax is. Since Gandalf doesn’t have facilities to transport fodder, he needs to graze him. A large horse in heavy work can need up to 6 stone of green fodder daily, if concentrated feeds are unavailable. Even if he’s carrying oats, S. will still need about 3 stone of grass or similar daily just to give him enough fibre to keep his gut working (constipation in horses is frequently fatal). It’s March when the ride takes place, so he’s unlikely to be able to cut significant amounts of green fodder to speed things up (horses can scarf cut grass or hay much faster than they can graze), and S. will still need time to digest after each meal before he starts moving again (this could be his sleep time). It’s quite likely that only half the time is being spent moving. So that gives an average moving speed of 12 mph. Allowing for a mixture of gaits to reduce horse and rider fatigue, that’s starting to permit significant sections at a hand-gallop. There may also be enforced sections at a walk either in darkness or where the road is in poor condition – they can’t risk even a minor injury to S’s legs – which will further increase the time spent in the faster paces. In fact, though, I think we’re told they’re travelling mostly by night anyway, and grazing/resting/lurking by daylight. This allows three-and-a-half nights’ worth of darkness (the half-night from the camp at the Isen, and three nights between Helm’s Deep and Minas Tirith) – say about 50 hours total time depending on latitude and weather – for them to be covering the 450 miles. Since only a fool would ride fast without moonlight when there’s no chance of getting a remount, that 9 mph average needs to allow for good chunks at the walk.

    Since we know the average stage of the beacons (and thus the errand riders) between Amon Din and Halifirien is 25-30 miles, we can suggest a section-speed for the post-riders of 1.5-2 hours (very dependent on the terrain – if the post stables are literally on the peaks with the beacons, the riders are stuffed!, whereas if the stables and barracks attached to each beacon are on the road and the guard has to climb up to the beacon each shift, the riders are much better off). This gives a total time from Minas Tirith to Rohan (borders) of 10-13 hours (I’m allowing for some time to get clear of the city at the beginning – Amon Din is outside the Rammas). If there is a similar system of post-horses in Rohan, total time for a message (a physical message, delivered in person) could be under a day, with a return message in a similar time-frame. However, it is often assumed that the party of riders G. meets on the night of the 7th includes the errand rider who reaches Theoden on the night evening of the 9th; this suggests that either they weren’t the same riders or the system broke down somewhere.

  39. #39 blf
    August 1, 2010

    What about the book? … It seems that Pippin doesn’t like the Beacon of Gondor.

    Yeah, this was one of the differences from the books which sortof irritated me about the movie. In this case, as I recall, it was Ok-ish in the context of the movie, that is not a big irritant, but combined with the lack of Red Arrow, an irritant nonetheless.

  40. #40 Michael Seery
    August 1, 2010

    Prof. Allain, as of now you have a free beer waiting for you in Santa Monica, California. Just let me know when you’re in town.

    Great post.

  41. #41 Xenobio
    August 3, 2010

    I was going to say something about the signal towers reaching only to the border of Gondor and the rider with the Red Arrow (token of distress) but Tim H and blf took care of it.

    Rohan was originally Gondor territory ceded to the Rohirrim – ethnic northerners, that’s why they’re blondies – about 500 years before LoTR in exchange for military assistance. That’s why it’s a big deal that Saruman was tying them up at Helm’s Deep and Isengard. When you come to think of it you wonder why the bad guys don’t have more cavalry…my guess is that horses can’t stand the smell of orc.

  42. #42 tresser
    August 4, 2010

    no need to guess, we were given maps in the books.

    map
    http://www.douglas.eckhart.btinternet.co.uk/maps-middle-earth-01.jpg

    source
    http://www.douglas.eckhart.btinternet.co.uk/maps.html

  43. #43 Tom Billings
    August 5, 2010

    @ Xenbio

    “When you come to think of it you wonder why the bad guys don’t have more cavalry…my guess is that horses can’t stand the smell of orc.”

    More likely it’s climate. Horses need well watered grasslands for raising them in the quantities required for a large Cavalry force. That’s why the Rohirrim were given “The Fields of Rohan”, and why in RL the Huns settled the Hungarian plain. While the “Southrons” came with their cavalry from far enough away that they were off the map, the maps of Middle Earth showed no well watered plains under the control of Mordor. Even the Southrons had fewer cavalry than the Witch King needed, as shown by their defeat by the Rohirrim. Also, if the Oliphants smell the same way Elephants do in RL, then horses would be often spooked by the smell, and would be dangerously out of control if not kept separate. Maybe that is why the Witch King brought up the Oliphants only *after* his cavalry were already defeated.

  44. #44 Mike Lorrey
    August 6, 2010

    SOLUTION:

    Since the origin and destination are more or less north and south of each other, but my maps of middle earth show the mountain range bends to the west of both locations, then the whole “night time in the middle” thing is because the path of the signals curves westward and the terminator line at the time the event happens is between the westernmost signal fire half way through, and the origin and destination which are located at a longitude further to the east, so it only appears that a night happened when actually the signal just followed a curved path into night time to the west then back eastward into day time further north.

    With this explanation, either this event happened in the early AM, at dawn, or else Middle Earth rotates opposite direction to that of Earth and the evening sun sets in the East…. which is the real problem at hand.

  45. #45 charonme
    August 6, 2010

    What do you guys mean by “night”? I don’t see any night in the video. A slightly darker environment can be explained by far more possibilities than just a “night” or a solar eclipse. Also, I sencond [25]. Also, you might consider that more than one beacon might be visible from some of the other beacons. Eg: beacon 1 lights, some time later beacon 2 is lit then a few seconds later beacon 3 is lit because they saw beacon 1 being lit.

  46. #46 Mike Maxwell
    August 6, 2010

    Re parens and kittens: I believe there is an HTML tag that will make a paren blink on and off. Let t-sub-b be the period of the blink, and let t-sub-c be the time it takes a kitten to turn from dead to alive and back to alive (when the paren is restored). I believe that t-sub-c >> t-sub-b. A blinking paren should therefore result in a kitten being somewhere between dead and alive; so you will have invented Schroedinger’s Kitten.

  47. #47 Mike L
    August 6, 2010

    I once calculated how much energy was radiated through the thermal exhaust port of the First Death Star. FYI, the thermal exhaust port is 2 meters in diameter, or about the size of a womp-rat.

  48. #48 sirdennisc
    August 7, 2010

    0.5 miles = about 800 metres (not 1800 as stated… divide subsequent totals by 10). And yes movie time, especially in montages, does not equal real time. For that matter how does elevation/decreased oxygen affect combustion and visibility?

  49. #49 Anna Tar
    August 7, 2010

    This scene is my favourite of the movie. Years of preparation and maintenance, heroic music, dramatic lighting (I’ll assume that’s all the ‘night time’ effect is) — all to transmit just 1 single bit of information.

    So: 1-bit, no error correction, no security: if lightning struck any mountain top beacon, wouldn’t Gondor & Rohan armies be in for a surprise meeting somewhere halfway?

  50. #50 chip
    August 7, 2010

    Does it rain and lightning above the treeline often? It is my impression that rain is for the lower areas of the world. Maybe not.

  51. #51 Michael Dorosh
    August 7, 2010

    “You have forgotton one clear and crucial element: for narrative structure, the period between lighting one beacon and the next may have been shrunk by the filmmakers.”

    Bingo – the filmmakers accelerated this for dramatic and visual effect.

  52. #52 simhedges
    August 7, 2010

    I can remember laughing to myself the first time I watched the movie, because of highly implausible beacon sequence.

    Some of the beacons were reasonably located (especially the first at Minas Tirith, and the third), but many were in unprotected areas, exposed to all elements, with guaranteed frosbite for the keepers and and at around cloud level (cloud at the wrong height for beacons that far apart would have killed the signal) – i.e. thousands of feet up, which is rather implausible from a supply level – how *did* they get those heavy logs up onto peaks that only mountain climbers could access?

    It also seemed highly unlikely that in all cases two prior beacons would be visible to those waiting to light a subsequent beacon – thus risking a single beacon failure stopping the signal. And how do they resupply at that height?

    Fewer, smaller, lower beacons in human-tolerable locations might actually have worked. And beacons with damp organic matter (grass, straw) or something else that would produce a lot of smoke. Smoke? Yes. Because although those beacons lit at night would benefit from the dry wood and clear flame shown in the movie, those flames are hard to see in daytime. The only reason in the movie for the implausible “night” is to make the beacons visible to the movie cameras in the long distance shots where multiple beacons are shown.

    The beacon sequence was, for many people, stirring and moving. But, for me, it was hokey and underthought.

    Still, better than the giant hamsters in The Two Towers!

  53. #53 Sam
    August 7, 2010

    I heard, (now mind you it’s only a rumor) that two of the signal watchers hooked up with a couple of Dunlending women and well…you know…turned into a real glitch in the system, the prior signal men had to carry up more wood and boy were they pissed!

  54. #54 Michael Martinez
    August 7, 2010

    In the book, Gandalf and Pippin hide by day and ride by night (a fact stated at the beginning of the 12th paragraph in the chapter “Minas Tirith” in THE RETURN OF THE KING) so all these assumptions about Shadowfax running continuously are erroneous (as far as the book is concerned). The movies condensed a lot of time and landscape to keep the action moving.

    The last beacon tower only extended to the border of Anorien and the East Fold so the distance the extended was less than 200 miles.

    “…the maps of Middle Earth showed no well watered plains under the control of Mordor.”

    That is completely wrong. Sauron controlled the well-watered plains north of Mordor, where horse/wain-using Easterlings had dwelt for centuries (in fact, it was one of those Easterling nations whose proximity had been the reason for Gondor’s inviting the Eotheod to migrate south to Calenardhon in the first place).

  55. #55 Craig
    August 7, 2010

    An excellent geeky science & Tolkien conversation, and for once I find I may have some significant insights to offer!

    I live in the Sierra Nevada mountains near Yosemite. I’m on an exposed hillside at an elevation of about 3300′, and I have a pretty good view of the snow capped mountains to the East, and some to the North East. After putzing around with Google Earth one day, I found all the mountain tops I can see from my property.

    In relatively clear weather, the farthest one I can see is Merced Peak near Yosemite Valley, which is about 40.5 miles away. I can see at least 17 peaks that are over 30 miles away and have peaks from 8742′ (Mt. Raymond) to 11,794 (Tower Peak), and at least 5 that are about 40 miles away. If I was on an exposed point with good visibility, I think it’s extremely feasible to have a view to signal peaks 40 miles away in each direction… and quite possibly farther. In fact, the only high peaks I can see ar almost entirely directly to the East, as the farther North or South I look, there are too many lower peaks that stand in the way of my seeing the highest peaks. Basically, I see the snowcapped peaks mostly just to the East, which makes sense.

    I also unfortunately have a fair amount of experience seeing fires burning across the landscape, though none on peaks. But when I say I’ve seen many fires, what I really mean is that I’ve seen the smoke. Unless they are very close, I’ve never seen the fires themselves other than glowing at night from large fires within 10 or 15 miles. Again, that visibility has a lot to do with the smoke, though.

    Here’s a good example in particular: Pilot Peak is 6000′, has a fire lookout tower, and is a little over 9 miles away – it’s quite prominent on a nearby high ridgeline. A couple times each year the sun hits the windows on the fire lookout just right, and it’s quite bright and easy to see, like a bright yellow star shining on the top of the mountain. But seeing a bonfire on the peak – it’d have to be exceedingly huge, I think. At 9 miles you can just barely make out the dots that are individual trees the are silhouetted on the ridge top – and I believe that these are probably pines about 50′ to 75′ tall. Even a 30′ tall bonfire would just look like an orange dot at night, but I do think that in relatively decent weather it’d be clear that it was a signal fire. I daylight it’s the smoke you’d see – especially on a peak. It’s hard to tell what you’d be able to see, but assuming that the watchmen were trained in what to look for – i.e., the fires were occasionally lit to test the system and train watchmen, I’d think that 10 miles would be a no-brainer, and 20 miles is feasible. Beyond that, I think it depends on the size and type of the fire, and how long you could keep it burning at a high intensity.

    All of that said, the top of Merced Peak, dominating the local landscape 40.4 miles to the East of me at a towering 11,784′, well, it looks mighty small to me… just a bump among snowcapped bumps. I can only imagine that an enormous bonfire on the peak (were that possible) would still probably be all but invisible on all but the clearest of nights, and that the smoke during the day would be blown Eastward and away from me with such vigor that it would probably require a telescope to see.

    FWIW, it seems pretty clear to me that signal peaks would never be high altitude ones for all matter of practical reasons – and they wouldn’t need to be. There are plenty of medium altitude ridges and peaks that would not only forward the signal to teh next signal peak, but be close enough to the foothills to warn more people all along the route.

  56. #56 CAF
    August 8, 2010

    Dude, It looks like you have WAYYY too much time on your hands!

  57. #57 Spencer
    August 8, 2010

    Perhaps the one at night was delayed because the guy who was supposed to be watching was snoozing/playing chess?

  58. #58 some guy
    August 8, 2010

    Noticed a little goof. 5 miles per 10 seconds = 1,800 mph, not 18,000 mph.

  59. #59 Big Red
    August 8, 2010

    Great stuff Professor. Its been my favorite secuence for years now. Beeing somewhat of a physic nerd I love thinking about stuff like this! Cheers from Norway….. Thanks also to TORN for the link.. :-)

  60. #60 Georgia
    August 8, 2010

    Interesting blog post; you should know this is the first time I have ever read anything about physics voluntarily. That being said, if this is still really stressing you out, you should invest in the revised edition of The Atlas of Middle Earth, by Karen Wynn Fonstad. It has maps of all regions of Middle Earth, and even roughly sketched plots of the paths taken by the company. I couldn’t find anything on the beacons, but a closer examination tonight might find a decent estimate of the distance between Edoras and Minas Tirith.

    (p.s. don’t underestimate a Dunedain ranger. Aragorn is part-elvish; that means his vision could potentially be as good as Superman’s)

  61. #61 Rhett Allain
    August 8, 2010

    @Some guy,

    Thanks. I corrected the post above.

  62. #62 Brenda
    August 9, 2010

    Let us refer to another author for a little help. The Bard invented a thing called telescoping time for plot lines purposes. This not something you need to rack your brain over, or loose faith in the consistency of Tolkien, which we have all come to depend upon. Then again I have to admit I have found myself wondering about this very topic .I found the whole thing thoroughly entertaining and appreciate your “skillz” !! You are a true Ringer.

  63. #63 Schirf
    August 9, 2010

    My theory is that the chain of signal fires has a built in delay. The first few locations are close together (~5-7 miles), and can be seen at a relatively short distance. Once the signal has reached the higher elevations, the watchmen wait till after dark and start the fire at an appointed time. This allows the watchers down the line to know when to look for the fire and to be ready to react, should a fire be seen. This explains the night scene.

    As far as resupply, maybe the elves, little affected by the weather, are put to work hauling wood up to the mountain tops? The could be another reason why they’re sailing to Valinor… to escape the chores of mortal men.

  64. #64 RedDutchPW
    August 10, 2010

    Ehm, shouldn’t that be how fast ARE the beacons??

    It’s still a great sequence in the movie even if it is a bit unlikely.

  65. #65 Wickedpt
    August 11, 2010

    The Encyclopedia of Arda has a very nice article on the original Beacons of Gondor. There where 7 of them (Amon Dîn, Eilenach, Nardol, Erelas, Min-Rimmon, Calenhad and Halifirien), spawning trough 150 miles.

    Here’s the article:

    http://www.glyphweb.com/arda/b/beaconsofgondor.html

  66. #66 Rendal
    August 11, 2010

    You have to remember you are dealing with movie time, which is always exaggerated one way or the other. They couldn’t actually show you how long they thought it might REALLY take the message to travel the beacons, because the movie would have been over before it got half way there. LOL

    Also, as for noticing the fire and relaying the message….you probably would have three people per station(one in case of sickness or sending a ground/foot message as needed). Also, these would have been people that were set there by the Lords of Gondor, trained and with orders, they would have seen it quickly I’m sure.

  67. #67 Span Hanna
    August 14, 2010

    One caution: don’t leap to conclusions about “screen time” having any relation to “real time”. Think of all the transition montages (as film-makers call them) between a protagonist’s great idea/decision and the moment, somewhere down the track, when the first problem appears. Movie audiences don’t have the patience to watch every step in the rags-to-riches metamorphosis.

  68. #68 sirdennisc
    September 4, 2010

    Not to flog a dead horse — though using the beacons was meant to save horses — but the correction you attribute to “someguy” was actually posted by me (#048) the day before!

  69. #69 uli
    March 10, 2012

    ha, splendid! thanks so much for this article – you (and everyone commenting here) have put my burning questions at ease. ever since this movie came out, what troubled me most was the signal wandering through the night – middle earth simple cannot be that vast! but – alas – the idea of a solar eclipse happening (like jason suggested) is simply brilliant!