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They orbit around the same thing, the Sun, but don't fly parallel. They all have the same orbit central point, however.

Diagrams use this as a way to line them all up at one fictional point in time, for the sake of showing at-a-glance measurements between planets, to a certain scale.

I remember reading that it didn't start out that way, but all of the collisions between orbiting bodies eventually caused a net rotation where there are fewer collisions and the whole system is the most stable.

They all orbit the sun in the same plane because the sun and the planets all formed from the same gas cloud. That gas cloud that they formed from had a particular angular momentum associated with it (i.e. on average, all the particles were spinning around in a particular direction). When the gas cloud collapsed to form the solar system, the sun and all the planets ended up spinning in that same direction because angular momentum is always conserved.

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I assume by "parallel" you mean like their orbits are all on the same plane. This is no accident or coincidence, it is a result of how they formed. The solar system and the sun itself formed from the collapse of a giant cloud of dust and gas, and that cloud had some amount of rotational momentum. Such momentum is conserved and like how a spinning skater will speed up as their mass is pulled in closer, the bits of the massive cloud of dust collected into planets and formed sustained orbits. Those orbits will all be on the same plane because that is the net result of the conserved momentum.

By parallel, I assume you mean on the same plane. You are kind of right, on diagrams they are exactly on the same plane because well you are looking at a 2D representation. In reality, they are not exactly on the same plane. That said, there are pretty close to it, compared to the Sun's equator, Mercury is only at 3 degrees of inclination, same with Venus, the Earth is at 7 degrees, Mars and Saturn at 5, Jupiter, Uranus and Neptune at 6. So they are all pretty close to be perfectly on the same plane. Why?

Well first of all the solar system started as a big cloud of particles. All of them were moving around in random direction. You know a figure skating spins? Well it show phenomenon, if you spin and you bring back your limb toward the center of your body, you will spin faster. Same thing happened to the solar system. As gravity brought the cloud of particle together, it started to spin faster.

Particles started to hit each other, which created a favored direction of spin. Aka most stuff was spinning in one direction, so everything else go bumped into, leaving just one main direction of spinning. You can see that happening in this demonstration video at around 2:30min.

Now you have a ball of particle spinning in one direction. The second analogy I will use is when you are spinning a pizza dough. I won't go into the physics detail of why this is happening, but basically the fact that the dough is spinning will stretch it around the axis of rotation. Same happened with the cloud of particles that was our early solar system. It stretch around the axis of rotation, bringing all the particles into roughly the same plane where all the particles eventually came together with gravity into the different planets.

If you've ever saw a professional make pizza bases by throwing them in the air, spinning, it's more or less like that.

More specifically, gravity is attractive in all directions. When the gas cloud the solar system firmed out of started spinning along a specific axis (because of how it was contracting), there was an outward force (the centrifugal force) counteracting gravity in one plane, but nothing perpendicular to that plane. So all the matter started falling into the centre, except where it couldn't, because the centrifugal force kept it in orbit around the forming sun. Repeat the process on smaller scale for planets, and that's why most of them rotate the same way too.

Yeah, anything that was on another plane was scrubbed long ago

This sounds like the ancient Italian myth of the creation of the universe - we're all a part of a giant pizza, our planet but a slice of pepperoni

Now I'm hungry

To add onto that the particles that weren't traveling on the same plane would bump into each other averaging out the variation of velocities outside of the rotating disc

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Very few diagrams of the solar system are strictly accurate for a reasons.

1. The relative size of the planets + the sun are just too different from each other. The sun is 285 times as wide as Mercury, the smallest of the 8 (It should be 9, I still love you Pluto) major planets.
2. The distances between them are massive and if drawn to scale would never fit on a single page while still allowing you to see all of the orbits.
3. Most diagrams are 2 dimensional representations and the planets actually orbit in 3 dimensions.

While the major planets (and many other solar system objects) orbit the sun in roughly the same disc they aren’t all doing so on the exact same plane.

Put another way of you were to look at our solar system from the “side” such at the earth stayed “flat” as it rotated around the sun, ie it never appeared to move up or down, the other planets would NOT do the same. As they orbited they would appear to go above and below the line drawn between the earth and the sun by as much as 7° in the case of Mercury. The Earth-Sun orbital plane is called the ecliptic and it’s one reference point we use when talking about orbits, because we live on Earth so it’s convenient to do so. But it’s not the ONLY reference plane you can use.

For example, you might already know that the Earth is tilted relative to its orbit around the Sun, but did you know the Sun is tilted too? I’d we drew our plane relative to the suns equator instead of the sun Earth orbit, earth would appear to move above and below this imaginary plane by more than 7°! In fact Mercury, the planet with the greatest variation relative to the Earth/Sun plane (the ecliptic) is the one that varies the least relative to the Suns equatorial plane.

There are other reference planes too. You can make one based on the orbit of any object around the sun. Or you can find the average of all the planets and weight it relative to their mass. That particular plane is the Laplace invariable plane.

So why don’t they all orbit on the same plane? Because planetary system formation is messy! Everything (well almost everything) in our solar system started out as big dust cloud. Not everything was moving in the same direction but overtime as objects collided and gravity pulled things together their average radial velocity ended up in roughly the same direction. But roughly in solar system scales leaves a bit of room for variation. Collisions over time perturbed the orbits (and rotational angles) of different planetary bodies.

Take Uranus for example. It’s spinning basically on its side (90°) relative to its orbit! That means one of its poles points roughly in the direction of the sun.

Meanwhile Venus is spinning backwards relative to the other planets, which probably means it got flipped over at one point due to one or more collisions.

And on top of all that, despite what it may look like the planets don’t orbit in perfect circles either but in ovals. Earths distance from the sun varies between 147 and 152 million km over one orbit (one year). Little Mercury varies between 48 and 70 million KM in its rapid orbit (88 earth days). And poor demoted Pluto has an even greater orbital variation over its epic 90,560 day “year” coming as close as 4.4 billion km and getting as far away as 7.4 billion km. Meaning during part of its orbit it’s actually closer than Neptune. This lasts about 20 years and the recent recent occurrence was from Feb 1979 to Feb 1999. Unfortunately you’ll have to wait for about 230 years for it to happen again, so if you were born after Feb 11, 1999 you missed it.