In my previous post on Gollum-physics, I considered different ways that Gollum could see in the dark. However, there is something else that I find interesting regarding Gollum. Gollum lives on an island in an underground lake, right? He lives on fish and stuff that he can steal from the goblins. How often would he have to eat? That is what I want to look at.
To the novice, this question seems impossible to approach. But if you sneak up on this question carefully, you might be able to get an estimate. This is the difference between an expert at BS (me) and normal people. I'm not afraid to try and answer this question.
Where would you start? First, why would Gollum need to eat at all? Well, he needs energy. We mammals (I am assuming Gollum is a mammal) eat and breath and drink water (and sometimes beer). Our bodies then take this stuff and use it to give energy. Gollum needs energy to move around and attack Bilbo, that is for sure. However, he also needs energy to keep his body temperature up to some reasonable amount. We all need to do this. However, Gollum is in an underground environment that is typically at some constant (and cold) temperature. If he doesn't eat, his body would eventually reach the same temperature as the stuff around him. I assume this would be bad. For humans, we need to keep our body temperature above 85°F to function at the non-dying level. Oh, the sunlight can help increase our body temperatures also (not sure how big of an effect this would be).
Assumptions
To start this problem, I need to get some values. I am going to do this two ways. I will makes some guesses, but then I will proceed in a symbolic manner. That way, at the end I should be able to get an expression for how often he needs to eat fish. This expression can then be recalculated with different starting values. Here are some things we need:
- Cave temperature - Tc. If I had to guess, I would put the temperature of the cave rock and water at 50°F (10°C). This is a complete guess. The only thing I have to go on is that Bilbo says the water is cold when he accidentally touches the lake.
- Gollum's operating body temperature - Tg. I have this feeling that hobbits are a lot like humans. Since Gollum used to be hobbit-like, he could have a hobbit-like body temperature. But he isn't a hobbit. He has lived in that mountain for quite a long time. Let's say his body temperature is 85°F (29°C)
- Gollum's mass - mg. Ok, let's assume a hobbit is about 1 meter tall. This is right around half the height of a human. What about the mass though? I would think this would be less than half. Actually, I have done this before with estimating the mass of The Hulk. I will skip the details and just give a value of about 20 kg. I think that is way too low, but I'm sticking with it for now.
- Heat capacity of Gollum - Cg. I guess this would be around that of water with a value of 4.19 J/(g*K).
- Amount of energy from consuming one average sized fish - Ef. I am just going to have to look this up. This random site puts 100 grams of raw fish at 427 KJoules. That means Ef = mf(4.27 KJ/g) where the mass of the fish is in grams. Let's just say the average underground lake fish has a mass of 500 grams. That means that 1 consumed fish would be 2.13 x 106 Joules. See, it isn't so scary to just make crazy guesses. It's fun.
That's enough assumptions for now - I suspect I might need a couple more things, but let's go on. Why would Gollum's temperature decrease (if he didn't eat and keep up the temperature)? He has to lose thermal energy. There are three basic interactions that could account for this energy loss.
- Conduction. This is an energy transfer between two objects in contact. Energy is transferred from the object with the higher temperature to the object with the lower temperature. For Gollum, this would mostly be an interaction between him and the air since he is warmer than the air. Since air has a fairly low heat capacity, I am going to pretend this heat conduction with the air is small (or smaller than other types of energy transfer). Oh, but what about the water? Clearly, if Gollum is going to hang around in the water he is going to lose energy much faster than in the air. I guess this is why he has a boat to get to his home island. Let's assume that Gollum stays dry.
- Evaporation. If you sweat or have water on your skin, this water can go from a liquid to a gas. This phase change takes energy and guess where the energy comes from. Yes, it comes from the body and results in a decrease in thermal energy and thus temperature. Ok, I already said Gollum stays dry. If we assume he also doesn't sweat (maybe he uses Secret antiperspirant) then we don't have to worry about evaporative cooling.
- Radiation. Everything radiates energy. The rate that energy leaves an object depends on the surface area of the object and the temperature. Hotter objects radiate energy at a faster rate. This is the energy transfer I am going to look at with Gollum. Why? Because it's something I can estimate. What about a room with objects all at the same temperature? Do they radiate energy also? Yes. However, they don't get cooler since they absorb radiated energy at the same time they are losing energy. They are in equilibrium.
What is the rate of energy an object radiates? One model for this energy radiation is from the Stefan-Boltzman Law. This says that the power radiated from an object would be:
In this model, σ is a constant with a value of 5.67 x 10-8 Watts/(m2K4) - so, you need to put the temperature in Kelvins instead of Celsius. T is the temperature of the body and Tc is the temperature of the surroundings. The constant e is the emissivity of the object which is essentially a measure of how much energy the body reflects. It turns out that humans have an emissivity of close to 1 (a perfect blackbody has an e of 1). Oh, the A is the surface area of the object.
It looks like I have everything except the surface area. Let's say that Gollum is about the shape of a cylinder with a height of 1 meter and a radius of 15 cm. I guess I should instead call these dimensions L and r so that they can be changed later. Let me say that Gollum's surface area is:
First, remember that a cylinder has two circles (on the top and the bottom) as well as the side. And what is k? It is just another constant I added in to the equation for some flavor. Clearly Gollum isn't actually a cylinder, right? The k is a constant to take this non-cylinder shape into account. I am going to leave this at a value of 1 for now. I suspect that if you took all the skin off Gollum and flattened it out, the area would be greater than that of a cylinder. However, a good portion of Gollum's skin-surface area could be radiating energy right back to other parts of the skin. Just think of the arms. If you hold your arms by your side, part of your arm skin is facing the sides of your body. Of course, if you huddle in a ball you can decrease your exposed surface area even more. So, like I said, I will go with a k value of 1.
I can now put all this stuff together to get an estimate for the radiated power.
Boom. That's it. I have estimated values for all of these quantities, so let me put them in. This gives a radiated power of 117 Watts. But what about fish? The key here is that Gollum wants to maintain a constant body temperature. Let me get this radiated power in a different unit. The common unit for power is Joules per second. Let me change this "consumed fish" per second.
What does this mean? Does this mean that Gollum is going to sit there and eat a tiny piece of fish every second? No. That just has to be the average rate that he eats fish. I assume that Gollum has some fat and stuff to store energy. At least he has a stomach so that the digestion of the fish takes some time. How many fish a day is this equivalent to? All I need to do is a unit conversion from seconds to days.
That seems like a lot of fish to find in an underground cave lake. What are the fish going to eat? Ok, I hear what you are saying. Gollum didn't live on fish alone. He stole food from the Goblins (or ate goblins). Let's say that just 25% of his food comes from fish. That would still be 1 fish a day (or about 500 grams of fish a day). Seems high.
But what about the fish? Wouldn't a fish have to eat something every day? Well, I am really just making up stuff here - I don't think I have ever seen fish in an underground lake. I have seen a cave crayfish before (way the hell back in a cave). There are two important points here. First, fish and crayfish are not warm blooded. They run their bodies at the same temperature as the water so they don't need to eat to stay warm. Second, the cave crayfish don't eat much. They just find stuff that filters through the ground into the cave. There's not much of this stuff to eat and that's why these cave crayfish are rare. Oh, I just made up the part about what these crayfish eat. I really don't know.
