I've wanted to work out a world where summers are extremely hot except for two weeks in the middle of summer. Winters would be extremely cold except in the dead of winter when there would be two weeks of spring like weather.
Can a planet have unusual weather changes due to its interaction with other astronomical bodies?
Sure. The Earth already sees these. Tidal forces can cause shifts in the major air currents causing weather events like rain, snow, etc. Because these forces are caused by movements tied closely to the revolution of the Earth, there's a day-night pattern to most of it, slightly offset because the Moon rises and sets on a different schedule than the Sun.
You might have a habitable moon of a gas giant orbiting its star at a distance that provides an Earth-like climate for the moon (let's say the star's a bit hotter but the gas giant orbits further out to compensate). The weather on this moon would be more closely tied to the orbit of the moon around the giant; when the moon's between the giant and the sun would be the planet's summer, and quite a hot one as the orbit of the moon around a large planetary body would also place the moon at perihelion. Then, the winter, when the moon's on the dark side of its gas giant, would be fairly cold. A one-Earth-year orbit of the moon around the gas giant would not be implausible especially if the gas giant takes ten to fifteen years to get around the star.
The interruption of this normal pattern for two weeks might be the orbit of another moon of the giant in the same plane, which causes a long-lasting total solar eclipse for two weeks. That would explain a cool period in summer. The warm period in winter would be harder to explain, perhaps an elliptical orbit around the gas giant which puts the world close enough to this heat-absorbing and radiating mass that it acts as a proxy sun during perigee.
Here's a 'simple' solution which doesn't require extremely complicated assumptions about the planetary constituents, etc.
So to have a cold winter and a warm summer you just have a eccentric orbit, so the gas giant orbits closer to the star in the summer and further during the winter.
Now for the cold 2 weeks in the summer. Let's make it easy and say there's another planet in a polar orbit, with the same period as our gas giant, every year, once a year, it pulls in between our gas giant and the sun until blocking some percentage of the sunlight for a few weeks.
Making a hot 2 weeks in winter is a lot harder. Heating a planet is unbelievably hard, because planets are big. You can't heat them through some friction processes because then the planets orbit would decay. In fact in space there's really only one thing that heats a planet - a star. So lets just say we have a binary star system. The second star can be smaller than the first, only just reaching enough mass to actually turn lighter elements into heavier elements. Let's put it in a pretty far away orbit as well, let's say double the distance of our planet to the sun. Certainly enough to affect the amount of energy the planet receives, but not enough to cause any significant tidal forces. Remember the temperature difference between the hot and cold extremes can be due to a small (1% variation in the relative position of the planet when at apogee and perigee).
These types of system are somewhat uncommon. Typically most bodies orbit in the same plane due to some rather complex physics. But it's easy to imagine a system where a rogue star was captured by the primary star's gravity.
The 4 seasons on Earth are caused by our tilt. This affects the angle at which the light from the sun hits us.
If another body were to affect this tilt because its orbit interacted with us, you get different patterns. If the orbits were set up in a way that they affect the tilt in a regular way then you could absolutely get odd weather patterns.
Heck, you don't even need another body involved if you want. You could have a very fast procession. That's the time it takes for earth to wobble on its tilt. This would affect the angle more often and cause even weirder seasons.
Besides seasons tho, any massive body could affect the planet enough to cause issues. Similar to how the moon affects our tides. The problem is that anything massive enough and close enough would likely cause much more severe problems than simply weather. Massive earthquakes and such.
A second star could cause very odd weather patterns by our standards also. 2 Summers, 2 Winters a single fall and spring and so on.
A highly elliptical orbit will give us a start, with a very hot summer and very cold winter. For this set-up, seasons will be worldwide, with distance from the sun being the determining factor instead of the tilt of the axis.
Now we need a way to warm up the planet when it is in apogee and to cool the planet when it is in perigee. Please note that these solutions may be closer to science-sounding rather than science-based.
To cool the summer: a moon of the planet is generally stable, but as the planet's perigee is approached, the heat and gravitational stresses is enough to trigger volcanic activity on the moon, which sprays out enough material (chemical make-up to be determined) which dims the sunlight enough to drop the temperature for a couple weeks until the moon settles down as the planet moves farther out.
To warm the winter: the planetary system includes a dust ring of volatile chemicals whose inner edge is at the same distance of the apogee of the planet. Chemical reactions between the atmosphere and the in falling dust generates enough heat in the atmosphere to mitigate the winter for a couple of weeks.
It occurred to me that part of the problem is answered by a circumbinary planet. In this configuration, the planet warms up as it approaches the two binary stars, but cools off when one star passes in front of the other.
Unfortunately, that doesn't explain the increase in temperature during winter. I could throw in a third star, maybe a red dwarf, that was also in a circumbinary orbit. Our intrepid planet would approach it just as it happens to be closer in its own orbit.
For this sort of arrangement to occur, there would have to be some kind of orbital resonance going on. Does anyone have the math and the desire to try that out?
Another possibility is that one of the binary stars (a white dwarf stealing mass from the other star) erupts in a supernova while the planet is furthest from the pair. This is a condition that might last just long enough for life and intelligence to arise on the planet and realize that the white dwarf is about to go kerblooey for good.
