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In velocity map imaging (photo-dissociation and photo-emission), the ejected particles form a newton sphere. I didn't really get the concept why it is called a "newton sphere" and also why at the poles the distribution is $\cos^2(\theta)$?

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I think what has happened here is a bit of terminology drift.

One of the many topics that Newton explored was how to understand rotational motion by examining the behaviors of a pair of masses bound together by a coupling force (a string). A key point of his analysis was that if the string is under tension, it means that the two masses must be rotating around a common center of mass (located somewhere along the string). His analysis amounts to an intriguing early exploration of the non relativity of rotation.

Now think about all of that in terms of velocity map imaging, where the ideas is to use lasers to disassociate two ions whose masses are roughly comparable (versus emitting a super-light electron), e.g. methyl chloride. You again have two masses, which are the two component ions. They are again bound together, this time by a combination of ionic and covalent bonding. And finally, the two ions can rotate around each other in various well-defined (quantized of course) vibration modes.

The incoming photon essentially snips that bond between the ions, allowing all sorts of interesting data to be collected as the two masses sail off in different directions.

So, I'm pretty sure that Newton got invoked into the terminology of velocity map imaging because he was the first one to do an in-depth classical analysis of the behaviors of two masses behaving like bound ions rotating around a common center of mass.

So where is the terminology drift? Well, Newton didn't actually call the two objects at the ends of the binding string masses, he called them spheres. So, his analysis is informally referred to as "Newton's spheres" -- note the plural! -- and from that, my suspicion is that the singular version of the phrase, "Newton's sphere," came to be interpreted as the sphere of products produced by clipping the string between two "Newton's spheres" (the bound ions).

Such odd little drifts are actually fairly common in science and technology. One of my favorites was in early personal computers, where people would refer to the central semiconductor memory of a small computer as its "core" (as in "central" or "fundamental") memory. But the term "core" actually originated in a completely different early storage technology where a "core" was a quite literal little disk of magnetic material with a hole in the middle.

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