I guess dragons love gold. In the recent trailers for The Hobbit: The Desolation of Smaug (YouTube clip) you can get a glimpse of all the gold Smaug has. Why does a dragon need this much gold? I don't get it. Maybe Smaug doesn't actually like gold but he just wants to keep it away from other people.
Whatever his motivation for hoarding, I am going to try to get an estimate for just how much gold is in the Lonely Mountain.
Scale
This is the hardest part. How big is the dragon? How big is a hobbit? We need some value here. My original plan was to look at a section of the trailer that shows gold pieces flying through the air. If I measure the acceleration of one piece of gold as it falls, I can assume that it has Earth-like acceleration of 9.8 m/s2. From this, I could find the scale of something in the scene.
Well, that didn't work out too well. Instead, I am just going with plain estimations. Here is a shot of Bilbo sliding down a pile of gold.
I previously estimated that a hobbit was about 1.3 meters tall. That would put this gold hill with a slope side at least 5 meters long inclined at 20 degrees above the horizontal. I think that actually, this gold pile might even be much larger than this estimate.
Volume of Gold
Assuming the floor under the pile is flat, this would be a cone of gold like this.
If I know the radius of the base circle and the height of the cone, then the volume would be:
Since I know the length of the slope and the angle, I can find both of the radius and the height.
Using these values, I get a gold-volume of 39.6 m3. I'm just going to guess that this is one of about 4 piles of gold. Yes, these are crazy guesses, but I have to start somewhere. That would give a total gold volume of about 158 cubic meters.
How Much Gold?
No one cares about the volume of gold. Instead we want to know how much it's worth. We humans on Earth (not Middle Earth) put a price on gold in terms of ounces. Right now, 1 ounce of gold is worth about $1,300 (US dollars). If I convert that to grams, that is $49.5 per gram.
Now, how do I get the mass of the gold? According to Wikipedia, gold has a density 19.3 grams/cm3. Maybe I should say this is 19300 kg/m3 since my volume is in cubic meters. Oh well. Actually, let me say the density of the gold is 17 g/cm3 to account for the air spaces between the coins. You could do a packing calculation for small discs, but this is just a guess.
With this density and volume, I get a mass of 2.69 x 106 kg. On the market, this would be worth 1.33 x 1011 US Dollars. That seems like it's a lot of money, but at least it is less than a trillion dollars (which is 1012 dollars). Actually, I think this is more gold than has been mined on Earth since 1880. Check out this plot from Wikipedia.
You could get the data from this graph and do a numerical integration - but let me just estimate. This is almost a straight line. The area under this curve would give the total production of gold. Since it's straight, it would be the area under a triangle. Putting in the values I get 1.75 x 105 tons or 1.59 x 108 kg. This is much MUCH smaller than the amount of gold I calculated.
Bonus link: Here is an interesting BBC post with estimates for the total amount of human-controlled gold in the world.
Your Complaints
I know you have some complaints here is my estimate of what you might say.
- The estimate of a cone-shaped pile of gold is bogus. Really, the gold is all messed up and scattered.
- Why would you say the floor is flat? It's inside a mountain. It's probably either a hill or a depression.
- Does all the treasure have to be made of gold? Don't forget, the Arkenstone is in there too.
- You make assumptions about Middle Earth based on Earth. These are not the same places.
- Lord of the Rings isn't actually real. You are wasting your time.
These are all valid points (except for the last one - I am not wasting my time). Although my estimations might be off for anyone one of these reasons, how much would each assumption change the result? I guess that with different values, the amount of gold would be within a factor of 2. Even if I am off by a factor of 10, that is still a whole lot of gold.
How about some homework?
- Derive the formula for the volume of a cone. Calculus required.
- Make some estimates of the uncertainty for each of the quantities above in the calculation. Use propagation of error to determine the uncertainty in the amount of gold. Here are some ideas about uncertainty.
- Suppose that there is only 1.59 x 108 kg of gold (like in circulation on Earth). How big of a cone pile would this make?
- Experimentally determine the packing ratio for cylindrical coins (that is the percent of space that is air in a blob of coins).
- Bonus question from Twitter: If Smaug brought half this gold into the lair after he moved in, how many trips would it take? You can estimate how much he can carry either with his feet or mouth.
Some final notes. It seems possible that Smaug could have a significant portion of all the gold in Middle Earth. What would the dwarves do with all that gold after Smaug was defeated? If there is a mistake in the above calculations, maybe it was on purpose.
