Ignoring the fact that the galaxy is spinning and moving through space too, how long will it take until all of the planets are in the same position that they are now, relative to the sun? IOW, if I mapped the location of the planets right now, when will the map be correct again?
Depends upon how accurate you want the prediction. But in general the answer is somewhere between never and impossible to know. Famously there are no known closed form solutions for only three bodies, and the Sun, Jupiter and Saturn are massive enough that we minimally need to predict where they will be as a three body question. The influence of these two on the smaller planets is enough that there is zero chance of being able to make any useful predictions. We could simulate the orbits, but errors accumulate, and it is unlikely that the results would be useful for long enough.
If you ignore interactions, and just assume each planet goes around in its own orbit, it comes down to how tight you want to specify the same position. For arbitrarily small accuracy, the time becomes arbitrarily big, reaching never as a pretty good approximation.
Someone who knows a lot more about astronomy than me [ETA while I was writing, e.g., @Francis_Vaughan] will be along soon with a more accurate answer, but it probably depends a lot on how specific you are about what the “same positions as now” means.
If you look at the problem as simplistically as possible, by treating the planets’ orbits as simple two-dimensional figures in the same plane, and multiply the periods of all their orbits, you get something like 334,747,200.63 years. But this is neglecting the fact that the orbits are ellipses, not circles, and that they precess, all at different rates, of course. It wouldn’t be too hard to add in the periods of their precessions to the orbital calculations, which would obviously yield a much larger number.
But this is also ignoring the fact that they are not all in the same plane. And it is also treating their motions as if they were clockwork, never varying. But as @Francis_Vaughan points out, there are also the effects of gravity and other forces that change the orbital periods, precession, and orbital planes. And the complications go on and on.
Within 1%? I wonder if you make that map again this time tomorrow…
One year on Mercury is 170 something Earth days, meaning after one Earth day Mercury has < 1% movement relative to the sun per day, and Mercury has the shortest orbit.
(I am also assuming by “map” you mean something like plotting points of planets on the celestial sphere, but made from the Sun instead of the Earth.)
Simplifying the solar system to abstract point masses, and ignoring external factors (as specified in the OP), does the Poincaré recurrence theorem rule out never?
I think the problem may be that instabilities are capable of ejecting a smaller planet from the solar system at greater than solar escape velocity. That becomes never. Otherwise I would imagine that indeed, never is ruled out.
That is if we take the solar system as an isolated entity. Ejection of a planet leaves it at the mercy of external gravitational forces, and if we take the galactic system as a whole, recurrence might not be ruled out even then. But by this time we are talking about the possibilities of ejection from the galaxy, and then we are subject to expansion of the universe. Depends upon how big a never you are willing to rule out.
It’s a rather long time. Wikipedia has a cite for the
scale of an estimated Poincaré recurrence time for the quantum state of a hypothetical box containing a black hole with the estimated mass of the entire Universe, observable or not, assuming Linde’s Chaotic Inflationary model with an inflaton whose mass is 10−6 Planck masses.[19]
It estimates that as 10^{10^{10^{3883775501690}}} seconds. It would be slightly less if we could limit ourselves to just the solar system, but I’m not sure how valid that is.
I would interpret this question as asking about crude alignments, not how long it takes for arbitrarily-precise recurrence of a chaotic system. For instance, the Voyager Earth-Jupiter-Saturn-Uranus-Neptune “Grand Tour” opportunity supposedly occurs every 175 years,
1% might be an overly stringent request for sane timespans.
If we say, take 1% to mean a cube around the planet of size 1% of its orbit’s radius (which seems a reasonable start) it still gets us problems. (Not specified if it is ±1% or a 1% box.)
As noted above, the orbits precess, and the eccentricity of many of the planets is such that the elongation of the orbit is more than 1%. Earth is currently about 3%. So precession may be enough to ruin our day. Since the OP explicitly ruled Pluto in, we have one planet with a pretty wild orbit to try to get recurrence for. The Earth’s orbit precesses with a period of 112,000 years.
At the other extreme, we could only demand that the angular locations in the sky as seen from the Sun recur within 1% as measured relative to fixed stars (which may need to be defined carefully as well.) I suspect the OP would be happy with that.
There is a website link that allows you to play with planetary orbits. Suppose that you could map our solar system exactly on the simulator and that the simulation ran in real-time. I’m wondering how long it would take for the simulation to get back to its initial setting.
For reasons that are too embarrassing to admit, my calculation of 334,747,200.63 years was wildly inaccurate on the low side. The actual number is 91,790,790,282,243,900,000,000, or 91.8 sextillion years. And that’s using the simplest assumptions: circular orbits in a single plane, unaffected by gravitational anomalies.
Given that the sun is expected to die out in a few billion years, ain’t gonna happen.
BTW, here’s a good reason not to count Pluto as a “real” planet:
The fact that its orbital plane is tilted 17 degrees to the earth’s, much more than any other planet, makes it pretty clear that it was not formed at the same time and in the same way as all the others. One of these things is not like the others, one of these things just isn’t the same.
And I thought I had heard there was a theory that it was a captured object from outside the solar system, but apparently not.
Like other members of the Kuiper belt, Pluto is thought to be a residual planetesimal; a component of the original protoplanetary disc around the Sun that failed to fully coalesce into a full-fledged planet. Most astronomers agree that Pluto owes its current position to a sudden migration undergone by Neptune early in the Solar System’s formation. As Neptune migrated outward, it approached the objects in the proto-Kuiper belt, setting one in orbit around itself (Triton), locking others into resonances, and knocking others into chaotic orbits.
Just curious, are you including Eris, Haumea, Makemake, Gonggong, Quaoar, Sedna, Orcus and Salacia in your list for planetary alignment? If not, why not?