On the question of motivation and evidence for dark matter, there are some illuminating (so to speak) answers on this site. I'd like to understand in detail the foundations of one of the lines of reasoning, as stated succinctly in this answer:
...stars within galaxies are moving faster than the escape velocity of the gravitational pull of the center of mass of the galaxy...
If I were to attempt to unpack this, I might say something like the following:
There a formula for escape velocity, based on on the classical laws of physics, which should be valid for measuring the rotation of a galaxy (ie, general relativity correction would not change the conclusion we are after?). $$ v = \sqrt(2GM/r) $$
However, there are other techniques for measuring
Mcan be measured indirectly via the mass-luminosity relationship
v(here the speed of the outermost stars in the galaxy) and
r(the radius of the galaxy) can be inferred, via the Doppler Effect, by measuring the galaxy's red-shift (and some other measurement, I suppose?).
The velocity formula does not add up when the measurements are plugged in. Therefore, we can assume that either one of the measurements is incorrect, or the escape velocity formula is incorrect.
The most likely candidate is that the measurement for mass is incorrect, specifically because the mass-luminosity relationship is invalid. The theory being that some mass doesn't interact with light.
Am I roughly correct that the above is the argument for the existence of dark matter? I'm looking for the building blocks of the argument, its foundation in terms of measurements and theories. (And I do understand that the case for dark matter lies in a convergence of several lines of evidence, but I'm attempting to wrap my arms around just this one for now.)