Adopted two-label gravitational model: the inverse-square pair potential with coupling parameters, the empirical mass identification, and the measured value and uncertainty of the Newtonian constant

Adopt the following gravitational model and empirical premises for the two-body part of this page, inside the force and interaction branches of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance and Restricted interaction models: pair decompositions, antisymmetry and centrality options, potential forces and the gravitational branch and the central-force class of Adopted central-force model class: static central pair potentials, the reduced one-label problem, the fixed-centre idealization, reading conventions and exclusions clause (i).

(i) The two-label gravitational model. A model of the gravitational branch is a model of class (i) of Adopted central-force model class: static central pair potentials, the reduced one-label problem, the fixed-centre idealization, reading conventions and exclusions with $J_0=(0,\mathrm{\infty })$, $Q=\{x=(x_1,x_2):x_1\mathrm{\ne }{x}_2\}$, masses $m_1,m_2>0$ in the equation of motion, and the gravitational pair potential $U(x_1,x_2)=-\frac{G_{\mathrm{N}}{m}_1^g{m}_2^g}{\mid x_1-x_2\mid },$ where $G_{\mathrm{N}}>0$ is the Newtonian constant of gravitation, a constant of dimension ${\mathrm{\ell }}^3{\mu }^{-1}{\tau }^{-2}$ read in ${\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$, and $m_1^g,m_2^g>0$ are the gravitational coupling parameters of the labels, primitive model data distinct from the inertial masses $m_1,m_2$ (Restricted interaction models: pair decompositions, antisymmetry and centrality options, potential forces and the gravitational branch clause (v), Dimensions and SI unit conventions of the non-relativistic classical mechanics framework). The corresponding central pair potential of class (i) is $u(\rho )=-\kappa \mathrm{/}\rho$ with $\kappa :=G_{\mathrm{N}}{m}_1^g{m}_2^g>0,\text{dimension }\mu {\mathrm{\ell }}^3{\tau }^{-2};$ it is of class $C^{\mathrm{\infty }}$ on $(0,\mathrm{\infty })$, the model has no external forces, and coincident configurations are excluded. The force is attractive, and its inverse-square form is an adopted law of this branch, not a consequence of the equation of motion.

(ii) The empirical mass identification. For the quantitative applications of this page, adopt in addition the identification $m_i^g=m_i(i=1,2)$ of the gravitational coupling parameters with the inertial masses. This is an empirical assumption, not part of clause (i) and not a consequence of the equation of motion (Restricted interaction models: pair decompositions, antisymmetry and centrality options, potential forces and the gravitational branch clause (vi)); it is the content of the weak equivalence principle for these labels, and it is adopted here only where a statement explicitly says so. When it is adopted, $\kappa =G_{\mathrm{N}}{m}_1{m}_2$, and every conclusion that uses it inherits its empirical status; statements that never invoke clause (ii) remain conditional on clause (i) alone, with $\kappa$ expressed through $m_1^g,m_2^g$.

(iii) The measured value of the constant. When a model of this page is used quantitatively, $G_{\mathrm{N}}$ may be assigned the CODATA 2022 recommended value $G_{\mathrm{N}}=6.67430(15)\times {10}^{-11}{\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2},$ with standard uncertainty $0.00015\times {10}^{-11}$ in the same unit and relative standard uncertainty $2.2\times {10}^{-5}$: a qualified empirical premise, not a definition and not an exact value. The recommendation is a least-squares value from 16 measurements whose mutual inconsistency led the CODATA Task Group to expand its uncertainty by a factor of $3.9$, and the 2022 value agrees with the 2018 value because the input data were unchanged. Any numerical conclusion that uses this assignment inherits the uncertainty of $G_{\mathrm{N}}$ together with the uncertainties and conditions of the other model data actually used; no such numerical conclusion is drawn by this postulate itself.

(iv) Scope and exclusions. The force equations in clause (i) define the formal Newtonian point-particle model. The words weak-field and non-relativistic specify when it is intended as an approximation to a relativistic gravitational description. Let $c:=299792458\mathrm{m}{\mathrm{s}}^{-1}$ be the speed of light in vacuum, used here only as an external comparison constant, not as a parameter in the Newtonian equations. For a configuration $x=(x_1,x_2)\in Q$ with $\rho (x):=\mid x_1-x_2\mid >0$, define the source potentials per unit gravitational coupling as scalar functions ${\mathrm{\Phi }}_1,{\mathrm{\Phi }}_2:Q\to \mathbb{R}$ by ${\mathrm{\Phi }}_1(x):=-\frac{G_{\mathrm{N}}{m}_2^g}{\rho (x)},{\mathrm{\Phi }}_2(x):=-\frac{G_{\mathrm{N}}{m}_1^g}{\rho (x)},$ each of dimension ${\mathrm{\ell }}^2{\tau }^{-2}$. Fix the inertial frame of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance clause (i), with its time and spatial coordinates, as the comparison frame for the intended relativistic approximation. For a state velocity $v=(v_1,v_2)\in ({\mathbb{R}}^3{)}^2$, where $v_i$ is the coordinate velocity of label $i$ in that frame, define the dimensionless parameters ${ϵ}_{\mathrm{\Phi }}(x):=\frac{\max \{\mid {\mathrm{\Phi }}_1(x)\mid ,\mid {\mathrm{\Phi }}_2(x)\mid \}}{c^2},{ϵ}_v(v):=\frac{\max \{\mid v_1{\mid }^2,\mid v_2{\mid }^2\}}{c^2}\mathrm{.}$ For a motion $x:J\to Q$, a trajectory interval $\mathcal{I}\subseteq J$ and a dimensionless tolerance $\eta \in (0,1)$ chosen small enough for the intended approximation, the weak-field, slow-motion regime in this comparison frame on that segment means ${ϵ}_{\mathrm{\Phi }}(x(t))\le \eta \text{and}{ϵ}_v(x^{\mathrm{\prime }}(t))\le \eta \text{for every }t\in \mathcal{I}\mathrm{.}$ Small relative speed alone does not meet the speed condition: adding a common large velocity leaves $v_1-v_2$ unchanged. The formal Newtonian equations retain their frame transformation rule, but this approximation criterion is assessed in the specified comparison coordinates, as in the cited Newtonian-limit discussion; it is not asserted in every boosted frame. The choice of $\eta$ is application-specific; these conditions are applicability criteria, not a proved sufficient condition for a relativistic error bound. This model supplies no relativistic correction estimate or error bound. When $G_{\mathrm{N}}$ is assigned the value in clause (iii), these inequalities inherit its standard uncertainty and any uncertainty in the other inputs; no regime assessment is made here. The model treats the two labels as point particles with fixed masses, ignores external bodies, rotation, deformation, mass loss, radiation and any non-gravitational force, and excludes coincident configurations. It is not asserted that any physical system is exactly of this form; in particular, the identification of clause (ii) is not a framework-wide identification, and the inverse-square law is not derived from the balance laws of the framework.

(v) Status. Clauses (i), (ii) and (iii) are adopted assumptions of the model and its empirical inputs; this item contains no proof and no measurement of its own. The theorems of this page that invoke the model are conditional on these clauses, and agreement of their predictions with observations would test the combined model and apparatus assumptions rather than prove them.

• What is primitive and what is derived. The inertial masses, the gravitational couplings, the constant $G_{\mathrm{N}}$ and the pair form are model data; the reduced mass $\mu =m_1{m}_2\mathrm{/}(m_1+m_2)$ is derived from the inertial masses, and $\kappa =G_{\mathrm{N}}{m}_1^g{m}_2^g$ (or $G_{\mathrm{N}}{m}_1{m}_2$ under clause (ii)) is the parameter that enters the central-force statements.

• Why the identification is separated. Momentum, angular momentum and energy balances and the relative reduction use only the inertial masses and the potential; the equality of gravitational and inertial parameters enters only through clause (ii), so its empirical content is not silently absorbed into the mechanical theorems.

• Why the constant is separated. $G_{\mathrm{N}}$ is measured, not exact; a page that replaces it by a number must say so and must carry the uncertainty. The symbolic theorems of this page use the symbol; only explicitly numerical applications would use clause (iii).