Comment by dreamcompiler | Hacker News Reader

Comment by dreamcompiler | Hacker News Reader

> Anyway, I think this is so cool because everything else on the solar system map is so regular and orderly, but not quasi-moons!

Nothing in the solar system is regular and orderly. It's chaotic because 3-body problems are almost always chaotic, and >3-body problems are always chaotic.

Our solar system is a hundred-body problem (or more depending how small a "body" you want to include).

So why then does our solar system mostly appear to be regular and orderly? Because of the time scales we care about, and because for the majority of bodies (not Zoozve!), only one other body produces the overwhelming majority of gravitational influence. Which means we can cheat and treat many very short-term predictions as 2-body problems.

Over a one-month timeframe we can predict the solar system analytically as a bunch of 2-body problems. Easy.

Over a few hundred years we can use numerical methods. Harder, but it mostly works well enough.

Over a million-year timeframe it's completely hopeless. We have no idea where Venus will be exactly a million years from today.

Edit: 10 million years is a better threshold for hopelessness. A million years is still somewhat predictable for some bodies.

https://en.m.wikipedia.org/wiki/Stability_of_the_Solar_Syste...

We must have some idea where the planets were back in the time of dinosaurs. I mean, Earth was still approximately the same distance from the Sun, and presumably still #3 right?

Probably the order of the planets was the same but we can't be sure of the eccentricity of their orbits back then. We do know Saturn probably didn't have rings back then.

"We have no idea where Venus will be exactly [ten] million years from today."

I'm trying to figure out what this is supposed to mean. Of course we don't know exactly, but also of course we have some idea.

Like, could Venus have had an eccentric orbit that extended further from the Sun than Earth or closer than Mercury? Could it have had a moon?

Sorry I should have been more specific. We can't predict the accurate position of Venus in its orbit ten million years from now, and it's possible its orbit will have a different shape then too. However it's unlikely that the orbit would be so eccentric that it would intersect with Earth's orbit. That kind of deviation becomes more likely when we talk about a billion years.

What's the definition of chaos here? I thought chaos implied the measurement error growing quickly during propagation, but here it looks like it's growing pretty slowly. In what way is this more chaotic than the movement of a single accelerating body (where the error grows linearly with time)?

Chaos requires nonlinearity. A gravitational body's motion affects every other body, which in turn affects the original body, etc. So you can't analyze the parts of the system independently and then just add up the results. A body's motion indirectly affects its own future motion. Spacetime tells matter how to move and matter tells spacetime how to bend, as Wheeler said.

In two-body problems, this infinite recursion converges to a closed-form solution, which means we can just write down the solution to the differential equations that describe the system's state and predict it at any time in the future.

Unfortunately there are no true two-body orbits in the universe. And with 3 and greater numbers of bodies that pleasant closed-form convergence doesn't happen. You get sensitive dependence to initial conditions (SDIC), with faster-than-linear divergence, and you can't solve the differential equations: You have to integrate them numerically.

The time horizon over which this divergence happens for planets is long relative to the time horizons we care about, but it's still SDIC and it's faster than linear. The same thing happens with the weather (the "butterfly effect") but there the time horizon for divergence (the Lyapunov time) is short enough that we humans can easily notice it.

https://en.m.wikipedia.org/wiki/Lyapunov_time

Understood, thanks!

Similar to Hurricanes but were we can't tell where they will be next week.

Exactly the same mathematical phenomenon (chaos) but the timescales are different.

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