The consequences and the mechanisms of a shift of the Earth away from the sun First premise: I have already made this question in the worldbuilding section, but since it is a very specific question many have recommended me to write in the astronomy section so here we are.

Second premise: I'm Italian so please excuse any mistake

So, I'm writing a story about a fast and cataclysmic climate change. Obviously there are some artistic licenses but I would like nothing in my story to be completely impossible, only (at most) very unlikely. 

The climate change that I would like to include in my story is very extreme, so I had to discard any terrestrial forcing to cause it (no change of ocean currents, supervolcanoes and methane eruptions) because they could not act fast enough and would not be so catastrophic (compared to what I imagine). So I focused on astronomical forcing. I excluded the impossibly forcing (for example, a magical and radical reduction in solar luminosity) and those that would be more catastrophic than the climate change itself (i.e. a planetary impact).

The last possible (though very very unlikely) option I have left is a dramatic change in the Earth's orbit. I have some knowledge in physics and astronomy so I did a personal search to figure out what could cause such an orbital change.The most plausible option would be to pass a rogue planet (or star) into the solar system, but since I want to complicate my life, I would like the orbital change to happen without any notice, which is quite impossible if it is the passage of a planet or a star in the solar system to cause it. 

The only plausible explanations I have found to fully satisfy my scenario are either the passage of a primordial black hole in the solar system (a black hole with Jupiter's mass would have a schwarzschild radius of 2.8 meters and, if it did not come in direct contact with some celestial body, it may take in the solar system without notice) or passing a high-density cluster of dark matter (assuming an self-interacting form of dark matter) near or inside the solar system. Both are immensely unlikely options and very speculative (primordial black holes are hypothetical objects and we still have no idea what the dark matter really is and how it behaves) but I do not think they are impossible in the beginning, or is it wrong?

Assuming I want to move the Earth 2x10^7 km (if I do not mistake at this distance the average temperature of the planet would be between 15 and 18 ° C lower than now) farther from the sun, could this change take place in less than a decade and the earth's orbit would remain stable or simply the planet would be expelled from the solar system?

I also wondered what effects this sudden and drastic lowering of the Earth's temperature would have on the weather (especially considering that the poles would cool down much faster than the equator) but I think it's a pretty OT question so I leave it as optional
It sounds more like world-building than astronomy to me, but since they punted it here, I'll give a crack.

There's a few ways you could approach this and stay within good science but making change relatively quick is difficult.  This might be possible if the planet is in a situation like mars, with no moon and between two more massive planets, so it's more vulnerable to orbital perturbations.   Mars axial tilt is believed to have changed quite a bit and fairly quickly.   While the total solar energy is constant, a variation in axial tilt can lead to a variation in ice buildup where, generally speaking a lower tilt is better for ice buildup, as the higher tilt leads to warmer summers and that melts ice more easily.   Mars axial tilt varies from 13 to 40 degrees, but that variation can take hundreds of thousands or millions of years, which is probably longer than you're looking for.

Nearly any orbital change, even a rapid one would probably take tens of thousands of years minimum.  

Geological changes are probably a little faster.  While it's still in the direction of theoretical, and I think you're right to avoid a large volcano or something like that because the effect from the volcanic ash would be greater than any long term effect from climate change.   

About 3 million years ago, the formation of the Isthmus of Panama is thought to have significantly changed ocean currents that cooled the Earth by a few degrees that lead to the Pleistocene epoch and ice age cycle.   You'd need to get creative to figure out a way for a land bridge like that to make a radical change, but it's possible.

Another method, if the oceans are shallow on your planet, you could have a valley like the Mediterranean valley about 5.3-5.6 million years ago, with a natural land bridge keeping the ocean water out.   That would be a hot, dry, sandy valley.   Then when the land bridge gives way the ocean quickly flows in.   If your valley is big enough the oceans could drop several feet and if you have land or small islands near the poles, the falling ocean level could increase polar land which could trigger ice formation.   Glaciers don't form over water, only land.   This is a bit of a stretch cause you'd need two geographical oddities, very shallow oceans over the poles and a very large dry valley ready to be flooded, but it could turn a planet towards a major ice age.   Ice tends to keep growing once it starts.   It's a natural feedback mechanism.

Probably the best method would be a passing rogue planet giving your planet a gravity assist.  You'd want a larger rogue planet and it would need to pass fairly close to your planet to tug your planet into a longer orbit.  You'd probably want a gas giant mass and as it passed, it would probably need to be visibly larger than our moon in the night sky.   

You mentioned 20 million km, or a 13% increase in distance and about a 22% drop in solar energy received.  You wouldn't need that much because of feedback mechanisms.   Ice ages can be triggered by as small as a 1% difference and for radical, rapid climate change, pushing your planet just 4-6% further away should be enough.

You'd need to make your planet's year longer, using Kepler's 3rd law, you'd use the 3/2 power rule, so 4% further would make the planet's year 6% longer or an additional 22 days to a 365 day year.

As to how fast that would make the planet cold.  That's hard to say.   You'd probably notice longer colder winters and frosting pretty much immediately, but for ice buildup and enormous glaciers you're still probably looking at at least a few hundred years, but that's kind of a ballpark guess.

You could do the same thing with a theoretical primordial black hole but they've kind of fallen out of fashion since it's believed they don't exist.  

If you wanted to get creative you could have the rogue planet pass in-between say your earth like planet and a mars like planet, pulling the Earth orbit more eccentric and longer and the mars planet shorter, possibly leading to the two planets having crossing or nearly crossing orbits - which could lead to all kinds of scenarios and predictions like a possible planetary collision in 6,000 years or something or a near pass and enormous tides.

It would also stir up the asteroids and comets in our system likely leading to an increase in impacts following the pass for the next 100 or more years.   

The rogue planet would create tides too if it appeared larger than the moon in the sky and maybe an earthquake or two, but wouldn't need to be a global disaster.    Rogue planets passing that close would be very rare, but not impossible.

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