A Couple Of Watch Guys Sitting Around Figuring Out How To Destroy The Solar System

We find out what happens if you shrink the Sun and Earth enough to fit them in the palm of your hand, and it's not good.

The other day, a colleague and fellow watch enthusiast and I were sitting at a bar after an event, and got to talking about a rather strange scenario. We were talking about perpetual calendars and got onto the subject of orbital mechanics, the perpetual calendar’s existence depending on the fact that the Earth’s orbit is not a whole number of mean 24 hour days. I don’t remember exactly how the subject came up but my colleague asked, “What would happen if you somehow shrunk the Sun and Earth down so they could fit in the palm of your hand?”

The answer at least in general should have in retrospect been obvious but it wasn’t until I woke up the next morning with a few hours to kill before catching a train back to New York, that I started to think it through. I had wondered fuzzily before falling asleep the night before whether or not the orbit of the Earth wouldn’t stay the same if the masses were the same but by the light of day I realized I’d forgotten something basic: conservation of angular momentum.

Angular momentum is the rotational inertia of a spinning body, or one that’s orbiting around another body. It’s “conserved” because unless there’s some external force that acts on the orbiting or rotating body, the angular momentum does not change; this is the rotational scenario of Newton’s law of motion: “An object at rest remains at rest, and an object in motion remains in motion at constant speed and in a straight line unless acted on by an unbalanced force.”

If angular momentum is conserved, then shrinking the Earth and Sun down to a small enough size to fit in the palm of your hand should make the rotational velocity of the Earth around the Sun greater – a lot greater. This is a giant-sized version of the usual analogy for conservation of angular momentum, which is a spinning figure skater – as the skater pulls in their arms, they start to spin faster and faster, since the angular momentum stays the same as the diameter of the system shrinks. This is clearer in the formula for angular momentum:

L = M x V x R

where L is angular momentum, M is the mass (of the Earth in this case) V is the radial velocity and R is the radius of the orbit. Being in a rush and not trusting myself to do the calculation correctly, I turned to ChatGPT.

I have many reservations about AI, most of them having to do with what you’d expect from a content creator – I find the whole idea of training an AI on material that’s under copyright highly suspect, for one thing. However, I’ve also found ChatGPT very helpful as a research tool – given reasonably clear prompts, it can return accurate information and seems to have a way of presenting information that’s very accessible. I started with the assumption that to be able to fit the Earth’s orbit into the palm of your hand it would have to be four inches in diameter.

Here’s what the AI had to say:

“The Earth's orbital velocity would increase to about 8.79 x 10 to the 16th power meters per second, or roughly 29.3% of the speed of light. This extremely high velocity arises because reducing the orbital radius so drastically (from 150 million kilometers down to just a few centimeters) requires a correspondingly extreme increase in speed to conserve angular momentum. Quite mind-boggling!”

The next natural question to arise is how much kinetic energy an Earth-mass object moving at 29.3% of the speed of light would have. You may be thinking, “well, at the very least trying to hold such a thing would blow your hand off” but the outcome as it happens is much worse to put it mildly.

The formula for kinetic energy is:

K.E. = 1/2 m v²

where kinetic energy is the product of an object’s mass M and velocity V. A massive object moving that fast is going to pack a punch. Here’s what the AI had to say:

“The kinetic energy of the miniature Earth moving at that speed would be approximately 2.31x10 to the 58th power joules. That's an absolutely astronomical amount of energy—many orders of magnitude greater than the total energy output of our Sun over its entire lifetime!”

This gets us into the realm of trying to understand very big numbers intuitively, which is basically impossible as our senses evolved to enable us to judge physically relevant sensory input, not numbers used to characterize things like the age of the universe, the diameter of a proton, or the kinetic energy of an Earth mass object moving at over a quarter the speed of light. One way to look at that number – 2.31x10 to the 58th power joules – is to compare it to the energy released by a supernova, which is, I have read, around 10 to the 44th power joules. It would be easy to think that this means that our tiny Earth has about a third more kinetic energy than a supernova.

However, that’s not how powers work. Ten to the power two is 100, but ten to the power 3 is 1000, so an increase in one order of magnitude means a number ten times bigger than the preceding number. Our tiny shrunken hypothetical Earth has fourteen orders of magnitude more energy than a supernova – that is 100,000,000,000,000, which I think means it has a trillion times the energy of a supernova, which is already incomprehensibly big. One of my favorite popular science writers is the cartoonist Randall Munro, the creator of xkcd, who did a wonderful explanation of supernovas in which he quoted a scientist who told him, “However big you think supernovae are, they're bigger than that.” To illustrate the point he asked, in a “What If?” story, “Which would look brighter to you, a supernova as far away as the Sun is from the Earth, or the detonation of a hydrogen bomb placed against your eyeball?” It turns out that the supernova would be brighter – by nine orders of magnitude.

The next question that occurred to me (what can I say, it was a busy morning) was to wonder how fast the Sun would be spinning if it were shrunk down to the size of a grain of sand. As it turns out the rotational speed of the Sun is quite a lot faster than I’d thought – one rotation takes just 25.4 days. Shrunk down to 0.5mm, which is roughly the size of an average sand grain, here’s what you get:

“If the Sun were shrunk down to the size of a grain of sand with a diameter of 0.5 mm, its new rotational velocity would be around 2.22×10 to the power 19 radians per second. This corresponds to a rotational period of approximately 2.83×10 to the power minus 19 seconds, which is an incredibly rapid spin—far beyond anything physically possible for real materials, but interesting to consider hypothetically!”

“The tangential velocity at the surface of the ‘grain of sand’ Sun would be approximately 5.55×10 to the power 15 meters per second, which is about 1.85 billion percent of the speed of light.”

The AI felt it necessary also to point out the obvious, which was that such a speed is physically impossible, although I sort of got that without being told.

Of course, the whole thing has to be taken in context – the context in this case being black holes. The Sun is not massive enough to collapse into a black hole on its own, but if you confine enough mass-energy in a small enough space you are going to get a singularity – there were people who worried that the Large Hadron Collider at CERN would produce enough energy to make a miniature black hole and destroy the Earth although if that were going to happen I think, hahaha, that we’d know by now.

The Sun will collapse into a black hole if it is shrunken to a size smaller than its Schwartzchild radius, which is about 3 kilometers. That is also the radius of the event horizon, so the whole shebang – tiny Earth, moving at a sizeable fraction of the speed of light; tiny Sun, rotating at a physically impossible speed, and person unfortunate enough to be holding both in the palm of their hand – would be inside the event horizon. As it turns out, it is actually possible for stable orbits to form inside the event horizon of a black hole – an object in such an orbit could never escape past the event horizon but it wouldn’t fall into the singularity either. So leaving aside the impossibility of the Sun’s rotation in this scenario, the orbit of the miniature Earth around the miniature Sun might actually be stable. The person holding both in the palm of their hand, however, would have no such protection and would be sucked into the singularity faster than you can say “spaghettified.”

Where this connects with watchmaking is of course in the angular momentum of a balance. A freesprung balance with adjustable timing weights follows exactly the same principle of conservation of angular momentum – move the weights closer to the center of the balance and the rate speeds up; move them away and the rate slows down. The interesting thing is that despite the incomprehensible difference in scale, both the balance and the Earth-Sun system follow the same laws of classical mechanics. Even a black hole is a classical object, general relativity being a classical theory built on the foundations of Newtonian classical mechanics, in which quantities can vary smoothly – unlike quantum mechanics, in which only specific values are allowed. If you wanted to you could even calculate the time dilation effect produced by the increase or decrease in rotational velocity of the balance – it would be trivially small but not nonexistent.

Hey, maybe I’ll ask ChatGPT.