A gravitational circular relative orbit checked on declared model data: $r_0=2\cdot {10}^6$m, $\mu r_0{\omega }^2=u^{\mathrm{\prime }}(r_0)=3\cdot {10}^{12}$N, $E=-3\cdot {10}^{18}$J and ${\omega }_{\mathrm{l}\mathrm{i}\mathrm{n}}=\omega ={10}^{-9}$s${}^{-1}$

Declare the gravitational two-label model of 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 with time window $I_0=\mathbb{R}$ and the same data as the barycentric check (Barycentric orbits of two unequal masses on declared model data: $p_1=(m_2\mathrm{/}M)p$, $p_2=(m_1\mathrm{/}M)p$, semiaxes in the ratio $3:1$ and areal rates in the ratio $9:1$): $m_1=2\times {10}^{24}\mathrm{k}\mathrm{g},m_2=6\times {10}^{24}\mathrm{k}\mathrm{g},\mu =\frac{3}{2}\times {10}^{24}\mathrm{k}\mathrm{g},G_{\mathrm{N}}={10}^{-24}{\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2},$ $\kappa =G_{\mathrm{N}}{m}_1{m}_2=1.2\times {10}^{25}\mathrm{k}\mathrm{g}{\mathrm{m}}^3{\mathrm{s}}^{-2},J=6\times {10}^{27}\mathrm{k}\mathrm{g}{\mathrm{m}}^2{\mathrm{s}}^{-1},$ with the couplings identified with the inertial masses. Then Circular relative orbits of the central model: the critical-point criterion $u^{\mathrm{\prime }}(r_0)=J^2\mathrm{/}(\mu r_0^3)$, uniform rotation of both labels about the centre of mass, and the linearised radial frequency gives:

1. the critical radius and angular velocity $r_0=\frac{J^2}{\mu \kappa }=2\times {10}^6\mathrm{m},\omega =\frac{J}{\mu r_0^2}={10}^{-9}{\mathrm{s}}^{-1},$ and the criterion holds in all three equivalent forms $(U_J^{\mathrm{e}\mathrm{f}\mathrm{f}}{)}^{\mathrm{\prime }}(r_0)=0$, $u^{\mathrm{\prime }}(r_0)=J^2\mathrm{/}(\mu r_0^3)=3\times {10}^{12}\mathrm{N}$, $\mu r_0{\omega }^2=3\times {10}^{12}\mathrm{N}$;
2. the orbital energy $E=U_J^{\mathrm{e}\mathrm{f}\mathrm{f}}(r_0)=-\mu {\kappa }^2\mathrm{/}(2J^2)=-3\times {10}^{18}\mathrm{J}$, the minimum of the effective potential;
3. the linearised radial frequency ${\omega }_{\mathrm{l}\mathrm{i}\mathrm{n}}=\sqrt{(U_J^{\mathrm{e}\mathrm{f}\mathrm{f}}{)}^{\mathrm{\prime }\mathrm{\prime }}(r_0)\mathrm{/}\mu }=\sqrt{\kappa \mathrm{/}(\mu r_0^3)}={10}^{-9}{\mathrm{s}}^{-1}$, equal to the orbital angular velocity $\omega$ for this inverse-square potential;
4. both labels move uniformly on circles about the centre of mass with angular velocity $\omega ={10}^{-9}{\mathrm{s}}^{-1}$ and radii $\frac{m_2}{M}{r}_0=1.5\times {10}^6\mathrm{m}$ and $\frac{m_1}{M}{r}_0=5\times {10}^5\mathrm{m}$, each with period $2\pi \mathrm{/}\omega =2\pi \times {10}^9\mathrm{s}$.

The stated period is relative to the centre of mass, for the coordinates $x_i-X$. An absolute label trajectory has this period only if the centre of mass velocity $V$ vanishes; otherwise the centre of mass drifts uniformly.

• Linearisation only. The equality ${\omega }_{\mathrm{l}\mathrm{i}\mathrm{n}}=\omega$ is a statement about the quadratic radial approximation; no nonlinear stability of the circular orbit is claimed.

• Declared data and the empirical status. The masses, $J$ and the declared $G_{\mathrm{N}}$ are model data chosen for exact arithmetic; the measured value and uncertainty of 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 clause (iii) are not used, and no observation is modelled.