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The supplementary question would also be about how the planets' orbits lie so close to a plane.

t takes almost 26,000 years for the axis to make a complete circle (to sweep out a cone).

..... and why was it a disc?The Sun would have been formed from an immense Nebula - a cloud of Hydrogen gas and dust which pulls itself together by the gravity due to its mass. This cloud would have been rotating a bit around some axis (being stationary would have been a real exception). As the Nebula collapsed, the angular momentum would have been conserved so the spin would have become faster and faster (like a skater, as she brings her leg and arms closer into her body).Smaller and smaller - faster and faster, until the diameter of the 'blob' of matter was small enough for it to become unstable. Then, some bits flew off (the future planets) until the new Sun became a stable size for its rotation speed. The flung-out bits would all be in or near a plane (round the Sun's equator) and they would also be swirling and spinning with their axes more or less parallel with the Sun's axis. Random motions and gravity will send them bumping amongst themselves will result in the bits gathering into lumps (the planets), each one spinning with an axis roughly parallel to the Sun's axis. But they may well end up with tilted axes - like Uranus, for example, which rotates at about 90degrees.Planets are a bit 'lumpy'; they are not of uniform composition. This means that they will probably wobble as they rotate. Large planets are molten inside and so the massive lumps can shift around inside them and, every now an then, the outside skin of the planet may move to preserve the total angular of the planet.Because they are not of uniform construction, they can interact with each other (gravity again) and one object's rotation can transfer to that of another object. This is why the Moon rotates at a rate which is the same as the rate that it orbits the Earth. It's also why the Earth's tides are causing the Moon's orbit to be increasing in diameter every year by a tiny bit.

It depends on whether you believe the universe is made from electromagnetic radiation.

It depends on whether you believe the universe is made from electromagnetic radiation. if you do then maxwells equations rule and the solution inside all spherical bodies are spherical harmonics governed by strattons equations. Whenever there is a join in the solution the equations change and produce different precession angles, hence the earth and mars solutions. The earth has one moon and mars has two. I can't help you much more than this you have to look at the maths.