From a post entitled “A Physicist Reads The Lord of the Rings”, written by a friend of mine who’s a big Tolkien fan:

For a moment the wizard stood looking at the fire; then he stooped and removed the ring the hearth with the tongs, and at once picked it up. Frodo gasped.

`It is quite cool,’ said Gandalf. `Take it!’

Exercise: Estimate (a lower bound for) the heat capacity of the ring. What other explanations can you think of for the fact that it remains `quite cool’?

Here’s the full excerpt from Chapter 2 of The Fellowship of the Ring:

To Frodo’s astonishment and distress the wizard threw it suddenly into the middle of a glowing corner of the fire. Frodo gave a cry and groped for the tongs; but Gandalf held him back.

‘Wait!’ he said in a commanding voice, giving Frodo a quick look from under his bristling brows.

No apparent change came over the ring. After a while Gandalf got up, closed the shutters outside the window, and drew the curtains. The room became dark and silent, though the clack of Sam’s shears, now nearer to the windows, could still be heard faintly from the garden. For a moment the wizard stood looking at the fire; then he stooped and removed the ring to the hearth with the tongs, and at once picked it up. Frodo gasped.

‘It is quite cool,’ said Gandalf. ‘Take it!’ Frodo received it on his shrinking palm, it seemed to have become thicker and heavier than ever.

The first question to ask: what temperatures are we talking about here?

Gandalf threw the ring “into the middle of a glowing corner of the fire”. Say the coals of the fire burn at roughly 1000 K, which is typical for a fireplace. (Steel is forged at around 1500 K, but it’s a cozy wood fire, not a forge.) The ring comes out of the fire at close to room temperature (say 300 K), and it’s been in there a “while”. That’s impressive for a small, metallic object. Here are some possibilities:

• The ring has an unusually high heat capacity.
• The ring has an unusually low thermal conductivity.
• The ring has some other mechanism for regulating its temperature.

We will start with the heat capacity hypothesis. Newton’s Law of Cooling can be written as

where \(T\) is the temperature of the object in \(K\), \(T_{env}\) is the temperature of the environment in \(K\), \(t\) is the time in seconds, and \(t_{\theta}\) is a time constant depending on the system with units of time \(s\).

This elementary differential equation can be solved as

where \(T_0\) is the temperature of the object at \(t = 0\) in \(\text{K}\). This means if we measure \(T_0\) and then later observe a temperature \(T\) at time \(t\), we can calculate the time constant as

For comparison, the time constant for an object with known surface area and heat capacity is

where \(c_p\) is the specific heat capacity in \(\text{J}/(\text{K}\, \text{kg})\), \(m\) is the mass of the object in \(\text{kg}\), \(h\) is the heat transfer coefficient in \(\text{W}/(\text{m}^2 \text{K})\), and \(A\) is the contact area available for heat transfer in \(\text{m}^2\).

For a metallic ring, the tricky part is finding a reasonable value for \(h\), since it varies by many order of magnitude depending on the system, but Wikipedia suggests:

\(h = 10\) to \(100 \, \text{W}/(\text{m}^2 \text{K})\) for air.

and since we are looking for a lower bound, we’ll take the lower value and assume that the heat transfer from the flames to the ring is like inefficient air. The ring has been known to change apparent weight and size, but let’s assume it has an effective area of roughly \(1 \, \mathrm{cm}^2\). (We don’t have to know the mass, although later we can make assumptions about the mass to compare heat capacities with other materials.)

Now we have to choose the time constant. We don’t know how long Gandalf left it in the fire, just that it was long enough to close the shutters and window. Since we are looking for a lower bound, let’s say it was closer to 100 seconds (≈1.7 minutes) than 1000 seconds (≈16 minutes).

Let’s say the ring starts at 300 K and after 100 seconds in the 1000 K fire increases in temperature by 1 Kelvin to 301 K. This would mean a time constant of

Using GNU units, this comes out to:

You have: 100 seconds / ln((300 - 1000)/(301 - 1000))
You want: hours
    * 19.430552
    / 0.051465341

or a little less than a day.

We conclude with a heat capacity of:

You have: (10 W / m^2 K) (19.4 hours) (1 cm^2)
You want: J/K
    * 69.84
    / 0.014318442

which is still about an order of magnitude larger than we would expect for something that small and metallic.

If the ring had the specific heat capacity of pure gold, it would weigh an uncomfortable \(540 \, \text{g}\) (more than a pound). With a relatively generous volume of \(0.5 \, \text{cm}^3\), this would require a remarkably high density, roughly 1100 g/cm³, with lead at only 11.34 g/cm³ and gold at only 19.3 g/cm³. (The densest natural element, osmium, is only slightly more dense, at 22.59 g/cm³. Still nowhere close to a dwarf star density of \(10^6 \, \text{g}/\text{cm}^3\), though.)

If the ring is not normal gold and has a mass of, say, 10 grams, it would require a specific heat capacity of \(7 \, \text{J}/(\text{g} \, \text{K})\), which is significantly higher than any known metal. Non-metallic crystalline solids aren’t significantly higher; their molar heat capacities are similar to metal since they tend to be roughly cubic and have only translational degrees of freedom.

Law of Dulong and Petit

The specific heat of copper is 0.093 cal/gm K (.389 J/gm K) and that of lead is only 0.031 cal/gm K(.13 J/gm K). Why are they so different? The difference is mainly because it is expressed as energy per unit mass; if you express it as energy per mole, they are very similar. It is in fact that similarity of the molar specific heats of metals which is the subject of the Law of Dulong and Petit. The similarity can be accounted for by applying equipartition of energy to the atoms of the solids.

From just the translational degrees of freedom you get 3kT/2 of energy per atom. Energy added to solids takes the form of atomic vibrations and that contributes three additional degrees of freedom and a total energy per atom of 3kT. The specific heat at constant volume should be just the rate of change with temperature (temperature derivative) of that energy.

http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/Dulong.html#c1

This means that high specific heat capacities tend to be correlated with low molar mass, like lithium and hydrogen.

Moreover, for these materials all but the aerogel will change substantially between \(300\, \text{K}\) and \(1000 \, \text{K}\), either melting, oxidizing, or undergoing a phase change. All but the aerogel will also get hot and stay hot.

Perhaps the ring has an unusual atomic structure which allows it many additional degrees of freedom, allowing it to soak up heat while still remaining a solid. As is apparent from the table, however, this would require an order of magnitude more degrees of freedom for anything with a large enough molar mass and density.

Perhaps the ring has an extremely low thermal conductivity, making it a kind of dense, lustrous aerogel. In this case, however it would not feel “cool to the touch” like a metal, but warm like a thermal insulator, more like a ring of glass or wood. It’s also not clear how to reconcile this with metallic appearance, since metallic sheens are associated with a continuous band, not a large band gap as in glass.

Of course, thus far we have been assuming that the ring’s mass is constant, when that is not necessarily the case. Frodo feels the ring become heavier as it nears Mordor. If the ring is somehow increasing its mass, it could increase its heat capacity arbitrarily, which could make it resistant to all but the highest of temperatures.

The main flaw in this argument is that the ring would have to become 50 fold denser, and should retain that 50 fold mass until it has cooled down by \(1 \, \text{K}\), which ought to take roughly the amount of time it spent in the fire, and which Frodo would have noticed as a major increase in weight, not a slight one.

Frodo received it on his shrinking palm: it seemed to have become thicker and heavier than ever.

But perhaps its change in density is not a predictable physical property, but something the ring can adjust at will to avoid detection. If we advance beyond known physics, we could speculate that the ring is transforming the heat energy itself into mass, incorporating that additional mass into its own substance.

Another difficulty with this hypothesis is that even with an adjustable mass, the surface of the ring would still get hot. This would mean that regardless of its actual temperature, it would not feel cool to the touch as metals feel. If the ring can adjust its own thermal conductivity, however, it could have a normal, metallic conductivity in most circumstances, but when heated it would decrease its thermal conductivity drastically, as a kind of negative feedback loop.

Then, when Gandalf removes it from the fire with tongs, the ring could detect its cooler environment and cool down its surface in the few seconds it lay on the hearth before Frodo picked it up.