Effective Mass of a Galaxy,
Star Velocity and their Relation

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60605

jodygeiger@informativity.org

ORCID: 0000-0001-5389-0447

ABSTRACT

We present a conditional effective-mass construction for a galaxy using an adopted mass-to-length count bound and a modelled velocity profile. The derivation distinguishes dimensionless count ratios from dimensional fundamental measures and relates the effective-mass parameter to a specified bound speed through a classical circular-speed mapping. We evaluate the fundamental measures, count rates and bound speeds using the 2022 Committee on Data for Science and Technology (CODATA) calibration with shared-input uncertainty propagation. The resulting fixed point establishes algebraic consistency under an imposed velocity equality. Comparison with a measured galactic rotation curve requires a separately specified source model and observational data.

Keywords: Effective Mass, Galaxy, Star Velocity, Count Constraint, Mass Bound, Measure

1. INTRODUCTION

We present an effective-mass construction for a galaxy using Measurement Quantization (MQ). The construction relates a specified modelled velocity profile to a bound speed through the count properties of measure.

The construction uses classical gravitational relations together with the MQ count constraint and the numerical-coordinate convention stated below. It does not fit a galactic rotation curve in this paper. Comparisons with approaches such as Loop Quantum Gravity [1, 2] and String Theory [3] concern different theoretical constructions; those approaches are not used in the calculation.

The fundamental length l_f, mass m_f and time t_f have International System of Units (SI) units m, kg and s ([4], Appx. W). Their dimensionless counts are n_L, n_M and n_T ([4], Appx. I, J). The exact speed of light is c=299792458 m s^-1 [5], with t_f=l_f/c. The radial length count is used for separation from the center of mass.

The effective mass M_e, in kg, is related to the bound mass M_b, in kg, by the velocity ratio in Eq. (1). The effective mass is an intermediate MQ parameter, and the bound mass is distinct from reconstructed baryonic mass. Here R is radial separation in m; v_m, v_b and v_e are the modelled, bound and effective speeds in m s⁻¹. Upright G denotes the positive upper-count gravitational coupling, in m³ kg⁻¹ s⁻². Upper-count refers to the asymptotic count limit. The mass parameter is obtained from

M_{b} = R v_{b}^{2} / G

through a classical circular-speed mapping; its interpretation for a galactic disk is qualified below at Eq. (13). Equation (1) requires R>0, v_b>0 and v_m≥v_b/2 for nonnegative mass. Table 1 reports the revised calibration and bound quantities.

M_{e} = \frac{R v_{b}^{2}}{G} (2 \frac{v_{m}}{v_{b}} -1) .

(1)

Within MQ, the Internal Frame is the discrete count domain of fundamental measures. The System Frame is its continuous relational encoding. The Frames mapping relates those discrete count configurations to their continuous representation (historical framework context in [4], Appx. K). Observable quantities are defined in the System Frame.

The dimensionless invariant coefficient θ_si has distinct angular and momentum realizations, Θ_si in rad and p_si in kg m s⁻¹ (historical angular and momentum descriptions in [4], Appx. A, Q, S). Their shared numerical coefficient under the MQ realization convention does not make angle and momentum the same physical quantity. Chen et al. [6] discuss conserved quantities in general relativity; that work does not derive the MQ realization convention. Within MQ, the metric differential is the representation mismatch when discrete count structure is encoded continuously under the Frames mapping; it is not itself a physical deformation. Its application retains the dimensions of the quantities being mapped (historical normalization context in [4], Appx. T).

The effective mass parameter enters the classical circular-speed relation used below. Its dependence on the specified radius and modelled speed is explicit in Eq. (1). This construction does not derive relativistic corrections from the separate relativity context [7], or establish quantum accuracy for a measured orbit. The count constraint supplies an upstream bound; its physical interpretation must be distinguished from the count rates evaluated in Eqs. (2) and (3).

The adopted modelled velocity v_m supplies mass-profile information. The kinetic-energy discussion in [4], Appx. AJ, does not uniquely derive the mass-scaling rule adopted below. The consequences of imposing equality of modelled and effective speeds are examined after Eq. (19).

2. METHODS

Earlier MQ applications are described in [8–13]. Here the discrete-count structure and count bound are model premises, rather than new empirical findings. The uncertainty-principle reference [14] supplies historical context; this paper does not derive integer counts of length, mass and time from it. The mass-to-length normalization, velocity input and shared calibration are specified explicitly.

The MQ framework and supporting arguments are described in Measurement Quantization [4]. We use its count structure as a hypothesis for this construction. Input-derived numerical agreement and internal algebraic closure do not constitute independent evidence that measure is discrete. The published source and its earlier preprint resources are identified in [4] and Online Resources.

https://www.researchgate.net/publication/361148724_OnlineResources

The source resources identify Suppl. 1 and Suppl. 2 as calculation spreadsheets associated with [4]. Published derivations and appendices are cited directly from [4]. These resources supply computational context rather than an additional observational dataset for the present paper.

3. RESULTS

For the radial application, R=n_Ll_f. Equations (2) and (3) evaluate the counts assigned to a one-second reference interval, expressed as rates. They do not identify arbitrary time and mass counts throughout a galaxy. The MQ calculation uses the 2022 CODATA inverse fine-structure input [15], retaining shared calibration dependencies. Parenthetical digits denote one conditional standard uncertainty in the final printed digits. We adopt the upper mass-count constraint discussed in [4], Appx. O, AF as a model premise. The reciprocal fundamental time specifies the corresponding count rate; evaluating that rate is not a derivation or independent physical validation of the bound.

\frac{n_{T}}{1 s} = \frac{1}{t_{f}} =1 . 85492188108 (14) \times 1 0^{43} s^{-1} .

(2)

The ratios of length, mass and time counts describe the count relations used in the construction. Their interpretation depends on the metric differential (historical normalization context in [4], Appx. T). Within MQ, the Informativity differential describes the count-dependent, non-Lorentz length contraction in realized separations, distinct from curvature and from the representation mismatch ([4], Appx. D). The mass-to-length ratio at the bound is specified below.

For the mass-count-rate calculation, Eq. (3) uses the mass rate m_f/t_f and divides it by the same m_f. Both quantities share the calibration input; their uncertainties are correlated. The resulting count rate is the reciprocal fundamental time, expressed per second. The upper-bound interpretation comes from the count constraint identified above.

\frac{n_{M}}{1 s} = \frac{4.03711234219 (62) \times 1 0^{35} kg s^{-1}}{m_{f}} =1 . 85492188108 (14) \times 1 0^{43} s^{-1} .

(3)

The rate identities alone do not determine how much galactic mass is observable. They evaluate a reference-interval count rate; the mass-to-length bound is a separate model premise.

For the galactic construction, the radial count describes separation relative to the center of mass. The escape-speed starting equation below uses the upper-count gravitational coupling. Its count ratio is dimensionless. At the stated bound, Eqs. (5) and (6) specify its numerical value using the fundamental-mass numerical coordinate in kilograms.

1. Calculations

We use the classical point-mass escape-speed identity for M=n_Mm_f, in kg, at separation R. Here v_escape denotes escape speed in m s⁻¹. Equation (4) assumes an isolated point mass and a gravitational potential that vanishes at infinity. It does not give the escape speed of an arbitrary extended galaxy, which depends on the full potential and its outer boundary [16].

The escape-speed relation has a factor of two under its square root. It serves only as the starting point for the count-based bound-speed construction in Eqs. (5)–(8). The later circular-speed inversion in Eq. (13) has a different role and is used without that factor.

v_{escape} = \sqrt{\frac{2G M}{R}} = \sqrt{2 \frac{l_{f}^{3}}{t_{f}^{3}} \frac{t_{f}}{m_{f}} \frac{n_{M} m_{f}}{n_{L} l_{f}}} = \sqrt{2 \frac{l_{f}^{2}}{t_{f}^{2}} \frac{n_{M}}{n_{L}}} = c \sqrt{\frac{2 n_{M}}{n_{L}}} .

(4)

Applying the adopted count-bound convention to the final form of Eq. (4) gives the unscaled bound speed below. The coefficient realization is applied separately, while the effective circular-speed mapping begins at Eq. (13).

In Eqs. (5)–(8), the ratio n_M/n_L is evaluated at the adopted mass-to-length count bound ([4], Appx. AF). It is the numerical count of fundamental mass units per fundamental length unit. Equation (6) makes the stipulated kilogram numerical-coordinate convention explicit by dividing the dimensional m_f by 1 kg. This ratio is dimensionless; the fundamental mass itself retains kg. The normalization is a model input, rather than an independent galactic mass measurement.

\frac{n_{M}}{n_{L}} =2 . 17643254057 (17) \times 1 0^{-8} .

(5)

Equation (6) fixes the numerical coordinate used in the speed calibration. This convention is additional to the reference-interval count-rate evaluations.

\frac{n_{M}}{n_{L}} = \frac{m_{f}}{1 kg} .

(6)

The same numerical count ratio enters Eq. (7). The symbol v_bc denotes the unscaled bound speed, in m s⁻¹; it precedes the coefficient realization used to obtain the bound speed in Eq. (8).

v_{b c} = c \sqrt{\frac{2 n_{M}}{n_{L}}} = c \sqrt{\frac{2 m_{f}}{1 kg}} .

(7)

Applying the dimensionless coefficient θ_si to the unscaled bound speed gives Eq. (8), under the metric differential convention ([4], Appx. AF). The value is an MQ calculation from the 2022 CODATA calibration [15], conditional on the numerical count convention and realization assumptions.

v_{b} = θ_{s i} c \sqrt{\frac{2 m_{f}}{1 kg}} =204 . 053582468 (23) km s^{-1} .

(8)

Using the adopted modelled velocity v_m as input, we calculate v_e. Imposing equality of the two is a separate condition examined at Eq. (20).

We next form the relative difference between the modelled speed and the bound speed calculated in Eq. (8). The dimensionless symbol

Δ_{f (v_{m})}

denotes the relative velocity difference in Eq. (9).

Δ_{f (v_{m})} = \frac{(v_{m} - v_{b})}{v_{b}} = \frac{v_{m}}{v_{b}} -1 .

(9)

We adopt the MQ mass-scaling relation

M_{e} / M_{b} = 1 + 2 Δ_{f (v_{m})}

, with the modelled speed providing the mass-profile input ([4], Appx. AH, AI). Substituting Eq. (9) gives Eq. (10), followed by the simplifications in Eqs. (11) and (12). The deviation is fractional, rather than a percentage multiplied by 100; its definition alone does not derive the mass-scaling relation.

M_{e} = M_{b} (2 Δ_{f (v_{m})} +1) = M_{b} (2 \frac{v_{m}}{v_{b}} -2+1) .

(10)

M_{e} = M_{b} (2 \frac{v_{m}}{v_{b}} -1) .

(11)

2 \frac{v_{m}}{v_{b}} - \frac{M_{e}}{M_{b}} =1 .

(12)

For a specified bound speed, Eq. (13) defines a mass parameter through the classical circular-speed inversion ([4], Appx. AG). Its interpretation as enclosed mass requires spherical Newtonian geometry; a galactic disk requires its actual gravitational field [17]. The equation uses the same upper-count coupling as Eq. (4), with the speed calibration supplied separately.

M_{b} = \frac{R v_{b}^{2}}{G} .

(13)

Substituting Eq. (13) into Eq. (12) gives Eq. (14), and solving for effective mass gives Eq. (15) ([4], Appx. AK). This is an algebraic construction from the specified speeds and radius.

2 \frac{v_{m}}{v_{b}} - \frac{G M_{e}}{R v_{b}^{2}} =1 .

(14)

M_{e} = \frac{R v_{b}^{2}}{G} (2 \frac{v_{m}}{v_{b}} -1) .

(15)

The effective-mass construction depends on radial separation R and the adopted modelled velocity profile. Table 1 supplies the bound speed and the mass-per-radius coefficient in Eq. (13), not a galactic crossover radius. The bound speed is a parameter of this construction, rather than a demonstrated universal ceiling on measured circular speeds.

For completeness, Eqs. (16)–(19) give the effective circular speed from effective mass. They use Eq. (13) directly, with the same gravitational coupling in numerator and denominator. The nonnegative square root is selected on the domain stated before Eq. (1). No independent numerical bound-speed calibration follows from this cancellation.

v_{e} = \sqrt{\frac{G M_{e}}{R}} = \sqrt{\frac{G M_{b}}{R} (2 \frac{v_{m}}{v_{b}} -1)} .

(16)

v_{e} = \sqrt{\frac{G}{R} (\frac{R v_{b}^{2}}{G}) (2 \frac{v_{m}}{v_{b}} -1)} .

(17)

v_{e} = \sqrt{v_{b}^{2} (2 \frac{v_{m}}{v_{b}} -1)} .

(18)

v_{e}^{2} =2 v_{m} v_{b} - v_{b}^{2} .

(19)

Imposing equality between the modelled and effective velocities reduces the velocity relation to the repeated-root fixed point in Eq. (20). At this point, the modelled, effective and bound velocities coincide. This establishes algebraic consistency within the stated construction. Establishing a minimum-energy orbit requires a separate constrained-energy derivation, while dynamical stability requires an analysis of perturbations about that orbit.

(v_{e} - v_{b})^{2} =0, v_{e} = v_{m} = v_{b} .

(20)

2. Revised Calculations

Table 1 presents the revised MQ calculations from the 2022 CODATA input [15]. The values share the same calibration input and retain its covariance. They supply calibration and bound quantities for the effective-mass construction.

Table 1. Revised MQ calculations using the 2022 CODATA calibration

Quantity	Symbol	Calculated value	Unit
Invariant coefficient	θ_si	3.26239030504(25)	1
Fundamental length	l_f	1.61619991148(12) × 10⁻³⁵	m
Fundamental mass	m_f	2.17643254057(17) × 10⁻⁸	kg
Fundamental time	t_f	5.39106261131(41) × 10⁻⁴⁴	s
Time-count rate	1/t_f	1.85492188108(14) × 10⁴³	s⁻¹
Mass rate	m_f/t_f	4.03711234219(62) × 10³⁵	kg s⁻¹
Count ratio at bound	n_M/n_L	2.17643254057(17) × 10⁻⁸	1
Upper-count gravitational coupling	G	6.6740779383(10) × 10⁻¹¹	m³ kg⁻¹ s⁻²
Unscaled bound speed	v_bc	62.5472624022(24)	km s⁻¹
Bound speed	v_b	204.053582468(23)	km s⁻¹
Bound mass per radius	M_b/R	6.2387441236(24) × 10²⁰	kg m⁻¹

Parenthetical digits denote one conditional standard uncertainty in the final printed digits. All rows share the 2022 inverse fine-structure calibration; the count-ratio and speed rows use the numerical convention in Eq. (6). The mass-per-radius row follows from Eq. (13).

For a specified radius and modelled speed, Eqs. (13) and (15) give the bound and effective masses. A numerical radial mass profile requires those galactic inputs. Comparisons with galactic source models [17, 18] remain a separate application of the construction.

Parenthetical digits denote one standard uncertainty propagated from the 2022 CODATA calibration input, retaining shared dependencies and covariance. These uncertainties are conditional on the stated MQ relations, realization assumptions and numerical count convention. Additional model discrepancy has not been quantified and is excluded from the reported uncertainties.

4. DISCUSSION

The distinction between fundamental reference units and physical discreteness is central to this construction. Planck’s historical work [19] provides the unit context; the MQ measure assumptions and supporting arguments are specified in [4] and Suppl. 1. Table 1 contains input-derived calibration and bound quantities. Neither a choice of reference units nor repeated use of the same calibration supplies an independent empirical test of those assumptions.

An empirical galaxy test would require a fixed source-mass model, a velocity profile predicted without using the target rotation data, and propagated source and observational uncertainties [17, 18]. If the modelled velocity is inferred from those same target velocities, transforming it through Eq. (15) is not an independent prediction. Table 1 does not perform that test. Extension to planetary or atomic systems, additional interactions and finite-count effects associated with the Informativity differential require separate analyses.

5. APPENDICES

APPENDIX A: TERMS

The entries below supplement the definitions already introduced in the text. Measurement Quantization [4] provides the underlying framework.

The conditional 2022 calibration gives the numerical coefficient 3.26239030504(25) [15]. The historical value 3.26239 rad appears in Table I of Shwartz and Harris [20] as a calculated signal/idler angle for a specified diamond configuration. The retained ±2 μrad is an author-adopted comparison tolerance, not that paper’s standard uncertainty. The related down-conversion [21] and wave-mixing [22] results do not furnish an independent calibration of the MQ coefficient for this construction.

The exact speed of light and the fundamental time relation are specified in Sec. 1 [5]. In a light-propagation count ratio, n_L=n_T gives the same speed; this equality does not equate the generic mass and length counts or determine their bound ratio. The propagation interval is positive.

The upper-count gravitational coupling used in the revised equations has the conditional 2022 value

G = 6.6740779383 (10) \times 10^{- 11} m^{3} k g^{- 1} s^{- 2}

[15]. Finite-count values refer to a specified count and are distinct from this upper-count limit. This is an MQ model output, distinct from the separately adjusted CODATA gravitational constant [15].

The fundamental measures introduced in Sec. 1 retain their dimensional units in every calculation. Their relation to conventional Planck units depends on the specified MQ calibration.

l_p, m_p and t_p are the conventional Planck units of length, mass and time, in m, kg and s, respectively.

The dimensionless counts introduced in Sec. 1 have distinct rate and bound-ratio uses in Eqs. (2)–(6).

The subscript L in Eq. (4) denotes radial separation from the center of mass. A trajectory-displacement count is a different use of the notation.

n_Lc is the dimensionless count of l_f used for a change in light position relative to the observer. It is separate from the radial count used in Eq. (4).

n_θ is the dimensionless System Frame count associated with the electromagnetic demarcation in this fine-structure calibration. Here electromagnetic demarcation denotes the fixed interaction count distance at which electromagnetic interactions are realized within MQ. The mixed brackets denote rounding to the nearest whole unit. The conditional 2022 argument gives n_θ=⌊84.6005456716/2⌉=42.

The mixed-bracket convention in the preceding entry is distinct from ordinary floor or ceiling alone. The stated argument is away from a half-unit tie.

Q_L is the dimensionless residual count contribution in the MQ right-triangle realization of separation measured in l_f. It enters the finite-count normalization in the System Frame.

2Q_Ln_L is the dimensionless normalization modifier used for the Informativity differential. Its finite-count role is distinct from the upper-count limit used in the gravitational expressions above.

The exact Planck constant is

h = 6.62607015 \times 10^{- 34} J s

[15]. With π the dimensionless circular constant, the conventional reduced Planck input in J s is ℏ=h/(2π). The MQ upper-count quantity is

ħ = 1.0545349844417293 (56) \times 10^{- 34} J s

under the same conditional calibration; it is a distinct model output and does not replace the exact SI definition. Its uncertainty is propagated from the calibration input.

m_e is electron mass, in kg.

a₀ is the Bohr radius, in m.

α_{f}^{- 1}

α_{p}^{- 1}

and

α_{c}^{- 1}

denote the dimensionless fundamental, Planck-like and classical inverse fine-structure forms. The fundamental form uses the Internal Frame; the remaining two use the System Frame and differ through the Informativity differential. The 2022 source input used here is

α_{c}^{- 1} = 137.035999177 (21)

[15].

STATEMENTS AND DECLARATIONS

The author has no relevant financial or non-financial interests to disclose.

The author has no competing interests to declare that are relevant to the content of this article.

DATA AVAILABILITY

Table 1 contains the revised calibration and bound calculations. The supporting numerical provenance and shared-input covariance record accompany this revision. Suppl. 1 and Suppl. 2 are the existing MQ resources identified in Sec. 2.

The source-model pointer is https://astroweb.case.edu/ssm/models/MW2018model.dat. It identifies a separate galactic model, not the data underlying Table 1. No radial comparison from that model is calculated here.

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Effective Mass of a Galaxy, Star Velocity and their Relation