—————————— read pre-print on research gate (new window) ——————————

• v_m is the observed or modelled velocity. Together v_m and v_b constitute the beta of a typical relativistic expression. When v_m is incorrect, the two values diverge.

• M_e is the effective mass which represents the gravitational mass measurable by an observer.
• M_b is the bound mass which represents the upper count bound of m_f that can be measured per t_f in a homogenous mass distribution.
• v_b is the bound velocity which represents that velocity when mass events exceeds the upper count bound due to measurement quantization.
• θ_si can be measured as the polarization angle of quantum entangled X-rays at the degenerate frequency of a maximal Bell state. As an angle θ_si=3.26239 rad ± 2 μrad; as a momentum θ_si=3.26239030392(48) kg m s^-1 and with respect to the Target Frame, θ_si has no units. The relation of angle and mass is mathematically demonstrated, as well, by No-Ping Chen, et. al.
• c is the speed of light which may also be written as c=n_Ll_f/n_Tt_f=299,792,458 m/s such that n_L=n_T=1 is physically significant.
• l_f, m_f and t_f are the fundamental measures, more precise expressions for Planck’s units – length, mass, and time – that consider the effects of length contraction associated with discrete measure.
• G is the gravitational constant, 6.6740779428(56) 10^-11m³kg^-1s^-2 such that its value considers the effects of length contraction associated with discrete measure at the upper count limit. Italicized G identifies a measure not at the limit (e.g., G=6.6738448362(53) 10^-11m³kg^-1s^-2 at the blackbody demarcation).
• n_Lr describes the count of l_f representative of the position of an observable with respect to the frame of a center of mass.
• n_Lc describes the count of l_f representative of a change in position of light measured with respect to the observer’s frame of reference.

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.