What causes matter to initially rotate/spin/orbit? All I can find is the statement that in space particles of dust/gas/matter contract into a spinning disk due to gravity (to form stars, solar systems, galaxies etc.), with no explanation as to why the spin began. I see a lot about the conservation of angular momentum, but these discussions all presume that the 'spin' already exists. What caused the spin in the first place? Shouldn’t gravity simply attract particles of dust, gas or matter together along a straight path till they collide, as a magnet does to a paper clip? The magnet does not make the paper clip revolve around it, and if I fall off of a building, I don’t spin around the earth. I fall in a straight path till I collide with the earth. What am I missing?
[Physics] What causes matter to initially rotate/spin/orbit
angular momentumcelestial-mechanicsnewtonian-gravityorbital-motion
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You're right that the Sun being 4.5 billion years old makes observations difficult. The Sun goes around the Galaxy about once every 225 million years, so since the Sun formed it has gone around the Galaxy perhaps 20 times. The trouble is that the Galaxy is not like the Solar System: stars don't go around on nice nearly circular orbits, everything is a bit jumbled. To give you an idea, here's an estimate of where the stars that are currently in the Solar neighbourhood have been during the last orbit of the Sun around the Galaxy:
As you can see, they span over 80,000 light years (that's basically the full width of the Milky Way) just 1 orbit ago, so a supernova remnant that was near the Sun 20 orbits ago could be virtually anywhere. We can't measure the age of the Sun or supernova remnants accurately enough to help either (as in being able to say "aha! a remnant with exactly the age of the Sun!").
It's similarly difficult to find stars that may have formed near the Sun. About the best we can do is to look for stars with similar age and chemical composition to the Sun, but at 4.5 billion years old, the accuracy of stellar ages isn't terribly good. Asking more broadly what star formation conditions were like in the Milky Way when the Sun formed is also a difficult question and a topic of current research, see for instance this reasonably current review.
A potentially more fruitful approach is to look for Sun-like stars that are forming now and see what conditions look like for their formation - potential triggering mechanisms, how many stars formed in a group, etc. But this only helps inform what the formation of the Sun would have looked like - it's likely the Milky Way was a significantly different beast 4.5 billion years ago in terms of gas supply, morphology, ISM conditions, etc.
The Sun certainly has angular momentum. You are correct that if two objects are of the same mass, the speed of rotation of the smaller denser object must be greater than the speed of the larger more diffuse object if both are to have the same angular momentum. The Sun is estimated to contain 99.86% of the total mass of the original nebula from which it formed, packed into a much smaller diameter.
Helioseismology has been used to probe the density, composition, and motion of the interior of the Sun. It's estimated that the Sun's angular momentum of rotation about its axis is $S = 1.92\times 10^{41}\ kg\cdot m^2\cdot s^{-1}$.
In addition to rotating on its axis, the Sun (and the entire Solar System) revolve around the center of the Milky Way Galaxy. One revolution takes one galactic year, which equals about 225 to 250 million terrestrial years. It's estimated that the speed of galactic revolution at our Sun's distance from the center of the galaxy is 1/1,300 the speed of light, c.
So the Sun has angular momentum from revolving around the galactic center, as well as from rotating around its own axis. The Sun's angular momentum of revolution around galactic center is part of the entire angular momentum of the Milky Way, just as angular momentum of the planets is part of the entire angular momentum of the Solar System.
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The Universe starts out as a nearly perfectly smooth distribution of matter with tiny perturbations to the density and velocity distributions across it. If you pick any region of this early Universe, it almost certainly has an angular momentum, or in other words it has a slow net spinning motion. As matter collapses under gravity, angular momentum is conserved and the slow spin of a large object becomes the fast spin of a more compact object. This is analogous to spinning on a chair with your arms out then pulling them in - you spin faster.