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Measurement Quantization (MQ) offers a parameter free approach to describing galactic orbital dynamics as a behavior constrained by bounds to measure. Expanding on MQ solutions to discrete gravity, dark energy, quantum entanglement, and a handful of constants such as the gravitational constant and Planck's constant, we find that the orbital motion of stars still follows Newton's expression, but the mass of a system is constrained by the count n_M of m_f that can be measured per increment of t_f. Specifically, that count is described by the Planck frequency. Importantly, this phenomenon must also be described relative to the non-discrete Target Frame of the universe, requiring a frame transform equal to the radial rate of expansion θ_si.

But, before discussing effective mass, we will briefly review the MQ approach to classical theory.

MQ is a physically significant nomenclature that separates the fundamental measures l_f, m_f and t_f from counts of those measures n_L, n_M and n_T. Using an MQ description of discrete gravity along with discrete expressions for Heisenberg's uncertainty principle, escape velocity and the speed of light, we resolve three properties of measure: discreteness, countability and in reference to three frames of reference.

Calculation of effective mass is a function of bounds to measure relative to other measures. We are very familiar with one of those bounds. Often described with respect to the speed-of-light, what we call in MQ - the length frequency - is the lower count bound of length per count of time (i.e., l_f / t_f). While we are accustom to referencing the measures, this is a frequency. References don't change in magnitude; their counts do (i.e., (n_L / n_T)=1).

Thus, the total mass of a galaxy can exceed the mass frequency bound in elapsed time but the gravitational pull of a galaxy is constrained to the effective mass per increment of time. In the graph above, the bound mass

is identified by the purple line. It does not take into account the effects of mass distribution within a galaxy. With the final expression - the red curve - we complete the description.

Any value below the bound mass M_b (purple) will exhibit a classical behavior. The result is a such that velocity appears to even out for all stars in excess of the quantization crossover. When mapped to describe star velocities we find a standard deviation of 1.394 km/s for stars out to 84,000 light-years from the Milky way core.

Finally, we note that the expression describing effective mass is approached is a constraining function. That is, you enter the modelled velocity to resolve the effective velocity. Naturally, it would be prefereable to have a function with source inputs of mass and distance v=(GM/r)^1/2, but this data is not easily resolved. We recognize the physical correlation and proceed with the understanding that the output is not an equality to the source data. Rather, when the effective and modelled velocities do not match, the curves diverge indicating a description that is not physically significant.