Two-body, central forces and gravitation — Examples

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 clause (i) with time window $I_0=\mathbb{R}$ and the following model data (decimal numbers declared for this computation; $G_{\mathrm{N}}=2.5\times {10}^{-25}{\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$ is a declared datum of the example, not the measured value of clause (iii), and the couplings are identified with the inertial masses as in clause (ii)): $m_1=m_2=2\times {10}^{24}\mathrm{k}\mathrm{g},M=4\times {10}^{24}\mathrm{k}\mathrm{g},\mu =\frac{m_1{m}_2}{M}={10}^{24}\mathrm{k}\mathrm{g},\kappa =G_{\mathrm{N}}{m}_1{m}_2={10}^{24}\mathrm{k}\mathrm{g}{\mathrm{m}}^3{\mathrm{s}}^{-2}\mathrm{.}$ Suppose a motion $x$ of this model has a relative orbit with angular momentum magnitude $J={10}^{27}\mathrm{k}\mathrm{g}{\mathrm{m}}^2{\mathrm{s}}^{-1}$, conic parameter $p=\frac{J^2}{\mu \kappa }={10}^6\mathrm{m},$ and eccentricity $e=\frac{3}{5}$ (the conic data of The inverse-square gravitational relative trajectory has the Kepler polar form: $\rho =p\mathrm{/}(1+e\cos (\theta -{\theta }_0))$, $E=(\mu {\kappa }^2\mathrm{/}2J^2)(e^2-1)$, elliptic completeness and barycentric scaled ellipses clause 1). Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), as in the cited conic and period items. Then the cited items predict, as exact consequences of these data:

1. the energy $E={\displaystyle \frac{\mu {\kappa }^2}{2J^2}}(e^2-1)=-{\displaystyle \frac{8}{25}}\times {10}^{18}\mathrm{J}=-3.2\times {10}^{17}\mathrm{J}$, and $E<0$ with $e<1$, consistent with the elliptic branch;
2. the semiaxes $a={\displaystyle \frac{p}{1-e^2}}={\displaystyle \frac{25}{16}}\times {10}^6\mathrm{m}$ and $b=a\sqrt{1-e^2}={\displaystyle \frac{5}{4}}\times {10}^6\mathrm{m}$, with $a=-\kappa \mathrm{/}(2E)$;
3. the relative period $T=2\pi \sqrt{{\displaystyle \frac{\mu a^3}{\kappa }}}=2\pi {\displaystyle \frac{\mu ab}{J}}={\displaystyle \frac{125\pi }{32}}\times {10}^9\mathrm{s}$, so $T^2={\displaystyle \frac{4{\pi }^2\mu a^3}{\kappa }}$;
4. the third-law form $T^2={\displaystyle \frac{4{\pi }^2{a}^3}{G_{\mathrm{N}}M}}$ of Kepler's third law for the gravitational two-label model: the relative period is $T=2\pi \mu ab\mathrm{/}J=2\pi \sqrt{\mu \mathrm{/}\kappa }{a}^{3\mathrm{/}2}$, so $T^2=4{\pi }^2{a}^3\mathrm{/}(G_{\mathrm{N}}M)$ under the mass identification, and both labels are periodic relative to the centre of mass clause 2 under the declared identification;
5. the areal rate $J\mathrm{/}(2\mu )=500{\mathrm{m}}^2{\mathrm{s}}^{-1}$ and the identity $\pi ab\mathrm{/}T=J\mathrm{/}(2\mu )$;
6. both labels trace ellipses with the centre of mass at a focus, the same eccentricity $e$, and semimajor axes $a_1=a_2=\frac{1}{2}a={\displaystyle \frac{25}{32}}\times {10}^6\mathrm{m}$ (equal masses), each periodic about the centre of mass with period $T$.

• Conditional on the existence of the motion. The check verifies the numerical consequences of the cited items for any motion with these conic data; it does not construct the motion, and the existence of the elliptic branch is not part of the computation.

• Declared data only. The numbers $m_1,m_2,J,p,e$ and the declared $G_{\mathrm{N}}$ are model data chosen for exact arithmetic; no measured constant is used, and no real system is modelled.

Declare the same model data as the companion elliptic check (An elliptic gravitational relative orbit checked on declared model data: $p={10}^6$m, $e=3\mathrm{/}5$, $a=25\cdot {10}^6\mathrm{/}16$m, $b=5\cdot {10}^6\mathrm{/}4$m, $E=-3.2\cdot {10}^{17}$J and $T=125\pi \cdot {10}^9\mathrm{/}32$s, with $T^2=4{\pi }^2{a}^3\mathrm{/}(G_{\mathrm{N}}M)$): 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 $m_1=m_2=2\times {10}^{24}\mathrm{k}\mathrm{g},\mu ={10}^{24}\mathrm{k}\mathrm{g},\kappa =G_{\mathrm{N}}{m}_1{m}_2={10}^{24}\mathrm{k}\mathrm{g}{\mathrm{m}}^3{\mathrm{s}}^{-2},J={10}^{27}\mathrm{k}\mathrm{g}{\mathrm{m}}^2{\mathrm{s}}^{-1},p=\frac{J^2}{\mu \kappa }={10}^6\mathrm{m},$ where $G_{\mathrm{N}}=2.5\times {10}^{-25}{\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$ is a declared datum of the computation and the couplings are identified with the inertial masses. Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), as in the cited conic item. On this data the formulas of The inverse-square gravitational relative trajectory has the Kepler polar form: $\rho =p\mathrm{/}(1+e\cos (\theta -{\theta }_0))$, $E=(\mu {\kappa }^2\mathrm{/}2J^2)(e^2-1)$, elliptic completeness and barycentric scaled ellipses give:

1. Parabolic branch. The energy $E=0$ corresponds to eccentricity $e=1$; the conic is $\rho =p\mathrm{/}(1+\cos (\theta -{\theta }_0))$, whose denominator ranges over $(0,2]$, and the periapsis is ${\rho }_{\min }(0)=p\mathrm{/}2=5\times {10}^5\mathrm{m}$, with $U_J^{\mathrm{e}\mathrm{f}\mathrm{f}}({\rho }_{\min }(0))=0=E$.
2. Hyperbolic branch. With eccentricity $e=2$ the energy is $E=\frac{\mu {\kappa }^2}{2J^2}(e^2-1)=\frac{3}{2}\times {10}^{18}\mathrm{J},$ the periapsis is ${\rho }_{\min }(E)=p\mathrm{/}(1+e)={10}^6\mathrm{/}3\mathrm{m}$, with $U_J^{\mathrm{e}\mathrm{f}\mathrm{f}}\left({10}^6\mathrm{/}3\right)=E$, and the orbit is the branch $\rho =\frac{p}{1+2\cos (\theta -{\theta }_0)}$ evaluated where $1+2\cos (\theta -{\theta }_0)>0$, that is $\mid \theta -{\theta }_0\mid <2\pi \mathrm{/}3$ modulo $2\pi$.
3. Consistency of the classification. In both cases $e\ge 1$ and the corresponding focus-directrix conic loci of The inverse-square gravitational relative trajectory has the Kepler polar form: $\rho =p\mathrm{/}(1+e\cos (\theta -{\theta }_0))$, $E=(\mu {\kappa }^2\mathrm{/}2J^2)(e^2-1)$, elliptic completeness and barycentric scaled ellipses are unbounded; the barrier of clause 2 gives $\rho \ge {\rho }_{\min }(E)>0$ along the motion, so the collision is excluded; no claim is made that the motion traverses the entire locus.

• Conditional on the existence of the motions. As in the companion elliptic check, the computations verify the formulas for any motion with the stated conic data; they do not construct the motions or settle their asymptotics.

• Declared data only. The numbers are model data chosen for exact arithmetic; no measured value, observation or scattering measurement is used.

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 $I_0=\mathbb{R}$, unequal inertial masses and a declared value of the constant, $m_1=2\times {10}^{24}\mathrm{k}\mathrm{g},m_2=6\times {10}^{24}\mathrm{k}\mathrm{g},M=8\times {10}^{24}\mathrm{k}\mathrm{g},\mu =\frac{m_1{m}_2}{M}=\frac{3}{2}\times {10}^{24}\mathrm{k}\mathrm{g},$ $G_{\mathrm{N}}={10}^{-24}{\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2},\kappa =G_{\mathrm{N}}{m}_1{m}_2=1.2\times {10}^{25}\mathrm{k}\mathrm{g}{\mathrm{m}}^3{\mathrm{s}}^{-2},$ with the couplings identified with the inertial masses, and suppose a motion has relative conic data $J=6\times {10}^{27}\mathrm{k}\mathrm{g}{\mathrm{m}}^2{\mathrm{s}}^{-1},p=\frac{J^2}{\mu \kappa }=2\times {10}^6\mathrm{m},e=\frac{1}{2}\mathrm{.}$ Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), as in the cited conic and period items. Then the cited items give:

1. the relative semiaxes $a=p\mathrm{/}(1-e^2)=\frac{8}{3}\times {10}^6\mathrm{m}$ and $b=a\sqrt{1-e^2}=\frac{4\sqrt{3}}{3}\times {10}^6\mathrm{m}$, energy $E=-\kappa \mathrm{/}(2a)=-\frac{9}{4}\times {10}^{18}\mathrm{J}$, and period $T=2\pi \sqrt{\mu a^3\mathrm{/}\kappa }=\frac{16\pi }{3\sqrt{3}}\times {10}^9\mathrm{s}$;
2. each label's orbit about the centre of mass is the scaled ellipse with parameters $p_1=\frac{m_2}{M}p=\frac{3}{2}\times {10}^6\mathrm{m},p_2=\frac{m_1}{M}p=\frac{1}{2}\times {10}^6\mathrm{m},$ the same eccentricity $e=\frac{1}{2}$, focus at the centre of mass and semimajor axes $a_1=\frac{3}{4}a=2\times {10}^6\mathrm{m}$, $a_2=\frac{1}{4}a=\frac{2}{3}\times {10}^6\mathrm{m}$;
3. the areal-velocity magnitudes about the centre of mass are in the ratio $(m_2\mathrm{/}{m}_1{)}^2=9$: $(\frac{m_2}{M}{)}^2\frac{J}{2\mu }=1125{\mathrm{m}}^2{\mathrm{s}}^{-1},(\frac{m_1}{M}{)}^2\frac{J}{2\mu }=125{\mathrm{m}}^2{\mathrm{s}}^{-1};$
4. both labels are periodic about the centre of mass with the common period $T$.

• The focus is the centre of mass, not the heavier body. Both barycentric conics have the centre of mass at a focus; the lighter label 1 has the larger orbit, in the ratio $m_2:m_1=3:1$ of the parameters and semiaxes.

• Declared data only. The masses, $J$, $p$, $e$ and the declared $G_{\mathrm{N}}$ are model data chosen for exact arithmetic; no measured constant or observation is used, and no real system is modelled.

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.