Well, they do orbit at random angles. The equator is determined by what angle the planet is orbiting at, so before the planet has any form of weather and is just a lava ball, the temperature is equal all over. After it begins to cool down and show weather patterns, the widest section parallel to the sun is hotter and the thinner sections farthest from the sun are colder, making distict arctic circles and equators.
The planets all orbit in roughly the same plane because the solar system originally formed from a flat disk of dust.
As for why the plane is horizontal rather than vertical… remember that there is no up or down in space, so the solar system could be depicted in any orientation in diagrams, etc and be just as valid. In fact, compared to the flat plane of the **Milky Way Galaxy**, the solar system’s orbital plane is cocked at a steep angle.
Most of these answers are flat out wrong or answer the wrong question.
When the solar system was being formed, the was a cloud of dust, gas, and maybe rocks/ice in space much much larger than the current orbital radius of Pluto. If you calculated the center of mass of this cloud of material, every particle had some amount of angular momentum or angular motion around this point. If you did a vector sum (so things orbiting in opposite directions cancel out proportional to mass) of the angular momentum of every particle, you would find that there was some arbitrary “net” axis of rotation for the entire cloud. Normal (at 90 degrees) to this axis is the net orbital plane for the majority of objects in the solar system.
Since according to physics, angular momentum like linear momentum or energy, must be conserved in the absence of an outside force, the cloud will maintain this “net” axis of rotation as things crash into each other and stick together forming planets and larger bodies.
You can think about this, as if two objects, one moving downward across the orbital plane hits another moving upward across it, the resulting new body formed from the collision will have the “up” and “down” components canceled out, leaving only the motion in the direction around the axis along the net orbital plane.
Visual Explanation: [MinutePhysics](https://www.youtube.com/watch?v=tmNXKqeUtJM/)
Even if the solar system started as a “roundish” cloud of dust, the stars and planets only form when there is some rotation in the dust cloud. Due to conservation of momentum (both linear and angular), random collisions will eventually cause the dust cloud to become a “flat” spinning disk about the axis of rotation. Due to gravitational attraction, any particularly large accretion of dust starts to aggregate more and more material eventually forming the planets and the star at the center. This is why planets typically all orbit in a plane around a star. (occasionally larger stray bodies may be gravitationally captured after the system forms and those bodies may revolve at different angles – many comets, for example, are NOT orbiting our sun on the same plane as the planets)
In many cases, there may be enough material to form more than one star and you get binary systems.
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