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Circular relative orbits of the central model: the critical-point criterion $u^{'} (r_{0}) = J^{2} / (μ r_{0}^{3})$ , uniform rotation of both labels about the centre of mass, and the linearised radial frequency

Physical scope

Circular relative motion in the adopted central model: the critical-point condition of the effective potential, stated equivalently as the balance $μ r_{0} ω^{2} = u^{'} (r_{0})$ between the central force and the circular motion; the explicit uniform rotation of the relative coordinate and of both labels about the centre of mass; the necessity of the condition for a constant separation; and the frequency of the linearised radial equation at a critical point satisfying the nondegeneracy condition $(U_{J}^{e f f})^{''} (r_{0}) > 0$ . The linearised clause is a statement about the quadratic approximation only; no nonlinear stability, no existence of circular orbits at arbitrary radii and no empirical claim is made.

Statement

Adopt the central-force model 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) with masses $m_{1}, m_{2} > 0$ , $M = m_{1} + m_{2}$ , reduced mass $μ = m_{1} m_{2} / M$ , and static central pair potential $u \in C^{2} (J_{0})$ , and time window $I_{0}$ . Fix $J > 0$ , an oriented orthonormal triad $(e_{1}, e_{2}, e_{3})$ , and a radius $r_{0} \in J_{0}$ , and put $U_{J}^{e f f} (ρ) : = u (ρ) + \frac{J^{2}}{2 μ ρ^{2}}, ω : = \frac{J}{μ r_{0}^{2}} .$ Then:

(Criterion.) The following are equivalent: $(U_{J}^{e f f})^{'} (r_{0}) = 0; u^{'} (r_{0}) = \frac{J^{2}}{μ r_{0}^{3}}; μ r_{0} ω^{2} = u^{'} (r_{0}) .$
(Existence and the two labels.) If the condition of clause 1 holds, then for every $θ_{0} \in R$ the curve $r^{\circ} (t) : = r_{0} (\cos (θ_{0} + ω t) e_{1} + \sin (θ_{0} + ω t) e_{2}) (t \in R)$ is a central-force trajectory for $(μ, u)$ defined on all of $R$ , with $∣ r^{\circ} ∣ \equiv r_{0}$ , angular momentum $j = μ r^{\circ} \times (r^{\circ})^{'} = J e_{3}$ and constant energy $E = U_{J}^{e f f} (r_{0})$ ; and for every interval $I \subseteq I_{0}$ with nonempty interior and affine centre-of-mass curve $X^{\circ} : I \to R^{3}$ with $(X^{\circ})^{''} = 0$ , the restrictions to $I$ reconstruct a motion of the two-label model (Two-body reduction of a static central pair-potential model: the relative coordinate is a central-force trajectory for the reduced mass, the centre of mass moves uniformly, and the reconstruction is a bijection clause 4) has both labels moving uniformly on circles about the centre of mass: $∣ x_{1} (t) - X^{\circ} (t) ∣ = \frac{m_{2}}{M} r_{0}, ∣ x_{2} (t) - X^{\circ} (t) ∣ = \frac{m_{1}}{M} r_{0},$ with angular velocity $ω$ about $X^{\circ}$ .
(Necessity.) Conversely, if a motion of the model has constant separation $∣ r ∣ \equiv r_{0}$ on a time interval $I$ with nonempty interior and nonzero constant angular momentum $j = J e_{3}$ , then $(U_{J}^{e f f})^{'} (r_{0}) = 0$ and its polar angular coordinate satisfies $θ^{'} \equiv ω$ ; the relative motion is the uniform rotation of clause 2 up to the initial value of $θ$ , and both labels move uniformly on the circles of clause 2 about the affine centre of mass.
(Linearised radial frequency.) If $(U_{J}^{e f f})^{'} (r_{0}) = 0$ and $(U_{J}^{e f f})^{''} (r_{0}) > 0$ , then $ω_{l i n} : = \sqrt{(U_{J}^{e f f})^{''} (r_{0}) / μ} > 0$ , and a twice differentiable $δ : I \to R$ on an interval $I$ with at least two points solves the linearised radial equation $μ δ^{''} = - (U_{J}^{e f f})^{''} (r_{0}) δ on I$ if and only if $δ (t) = A \cos (ω_{l i n} (t - t_{0})) + B \sin (ω_{l i n} (t - t_{0}))$ for some $A, B \in R$ and $t_{0} \in I$ . This is a statement about the quadratic radial approximation at $r_{0}$ ; it does not assert nonlinear stability of the circular orbit.

Facts & Assumptions

Given: The central-force model of class (i) with the $C^{2}$ potential $u$ on the open interval $J_{0} \subseteq (0, \infty)$ , the data $J > 0$ , the oriented orthonormal triad $(e_{1}, e_{2}, e_{3})$ , the radius $r_{0} \in J_{0}$ , and the functions $U_{J}^{e f f}$ and constant $ω$ of the statement.

[L1]

Reduction and attached objects (Two-body reduction of a static central pair-potential model: the relative coordinate is a central-force trajectory for the reduced mass, the centre of mass moves uniformly, and the reconstruction is a bijection clauses 3-4, Central-force trajectories: the central-force equation, angular momentum, energy and the effective potential, Adopted central-force model class: static central pair potentials, the reduced one-label problem, the fixed-centre idealization, reading conventions and exclusions clause (ii)): the relative coordinate of every motion is a central-force trajectory for $(μ, u)$ ; conversely every central-force trajectory for $(μ, u)$ together with an affine centre-of-mass curve reconstructs a motion of the model, with $x_{1} - X^{\circ} = \frac{m_{2}}{M} r$ and $x_{2} - X^{\circ} = - \frac{m_{1}}{M} r$ .

[L2]

Mathematical supplier (Circular orbits of a non-radial central-force trajectory: the critical-point condition and the squared frequency of the linearised radial equation clauses 1-3, Oriented orthonormal triads in $R^{3}$ : existence over any unit vector, and the orthonormal expansion with no fourth orthogonal direction, Solutions of $v^{''} + ω^{2} v = 0$ are exactly the functions $A \cos (ω (t - t_{0})) + B \sin (ω (t - t_{0}))$ , Central-force trajectories: the central-force equation, angular momentum, energy and the effective potential): for a central-force trajectory for $(μ, u)$ with constant $j = J e_{3}$ , $J > 0$ , and $r_{0} \in J_{0}$ , (i) if $ρ \equiv r_{0}$ then $(U_{J}^{e f f})^{'} (r_{0}) = 0$ and $θ^{'} \equiv J / (μ r_{0}^{2})$ , the trajectory being $r_{0} (\cos θ e_{1} + \sin θ e_{2})$ with $θ (t) = θ_{0} + ω t$ ; (ii) if $(U_{J}^{e f f})^{'} (r_{0}) = 0$ then for every $θ_{0}$ the displayed $r^{\circ}$ is a central-force trajectory on $R$ with $j = J e_{3}$ and $E = U_{J}^{e f f} (r_{0})$ ; (iii) if $(U_{J}^{e f f})^{'} (r_{0}) = 0$ and $(U_{J}^{e f f})^{''} (r_{0}) > 0$ , the solutions of the linearised equation $μ δ^{''} = - (U_{J}^{e f f})^{''} (r_{0}) δ$ on an interval with at least two points are exactly $A \cos (ω_{l i n} (t - t_{0})) + B \sin (ω_{l i n} (t - t_{0}))$ with $ω_{l i n} = \sqrt{(U_{J}^{e f f})^{''} (r_{0}) / μ}$ .

[L3]

Differentiation and trigonometry (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , $C^{k}$ Euclidean maps and diffeomorphisms): derivatives of the displayed trigonometric curves and of the potential expressions are computed by the chain rule and the algebra of derivatives; the barrier term $ρ \mapsto J^{2} / (2 μ ρ^{2})$ has derivative $- J^{2} / (μ ρ^{3})$ on $(0, \infty)$ .

[L4]

Intervals and dimensions: $J_{0} \subseteq (0, \infty)$ is open, $r_{0} > 0$ , and $J > 0$ (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length).

Proof

technique · direct

1.1givenL3L4algebra

Clause 1: differentiating the definition of $U_{J}^{e f f}$ with [L3] gives $(U_{J}^{e f f})^{'} (r_{0}) = u^{'} (r_{0}) - J^{2} / (μ r_{0}^{3})$ , so the first two conditions are equivalent; and $μ r_{0} ω^{2} = μ r_{0} \cdot J^{2} / (μ^{2} r_{0}^{4}) = J^{2} / (μ r_{0}^{3})$ by the definition of $ω$ , so $u^{'} (r_{0}) = J^{2} / (μ r_{0}^{3})$ is equivalent to $μ r_{0} ω^{2} = u^{'} (r_{0})$ .

2.1step 1.1L1L2algebra

Clause 2: assume $(U_{J}^{e f f})^{'} (r_{0}) = 0$ . By [L2] clause (ii) the displayed curve $r^{\circ}$ is a central-force trajectory for $(μ, u)$ on $R$ with $∣ r^{\circ} ∣ \equiv r_{0}$ , $j = J e_{3}$ and $E = U_{J}^{e f f} (r_{0})$ ; restricted to any interval $I \subseteq I_{0}$ with nonempty interior and combined with an affine $X^{\circ} : I \to R^{3}$ , [L1] clause 4 reconstructs a motion on $I$ of the two-label model with $x_{1} - X^{\circ} = \frac{m_{2}}{M} r^{\circ}$ and $x_{2} - X^{\circ} = - \frac{m_{1}}{M} r^{\circ}$ . Taking norms gives the displayed constant radii; and since $(r^{\circ})^{'}$ is tangent to the circle $∣ y ∣ = r_{0}$ and $∣ (r^{\circ})^{'} ∣ = r_{0} ω$ , each label traverses its circle with angular velocity $ω$ .

2.2step 1.1L1L2algebra

Clause 3: assume $∣ r ∣ \equiv r_{0}$ on $I$ with $j = J e_{3}$ , $J > 0$ . By [L1] and [L2] clause (i), $(U_{J}^{e f f})^{'} (r_{0}) = 0$ , the polar angular coordinate satisfies $θ^{'} \equiv J / (μ r_{0}^{2}) = ω$ , so $θ (t) = θ_{0} + ω t$ on $I$ up to the additive constant, and $r = r_{0} (\cos θ e_{1} + \sin θ e_{2})$ is the uniform rotation of clause 2; the reconstruction of [L1] clause 4 gives the two labels on the circles of clause 2 about the affine centre of mass.

3.1step 1.1step 2.1L2L4algebra∎

Clause 4: under the two derivatives hypotheses, [L2] clause (iii) gives exactly the stated solutions of the linearised equation with the stated positive frequency $ω_{l i n} = \sqrt{(U_{J}^{e f f})^{''} (r_{0}) / μ}$ , as an identity on any interval with at least two points containing $t_{0}$ .

Remarks

The criterion is a force balance. The condition $μ r_{0} ω^{2} = u^{'} (r_{0})$ says that the effective central force at $r_{0}$ provides exactly the centripetal acceleration of uniform circular motion with angular velocity $ω$ ; no circular orbit exists at a radius where the effective potential has no critical point.
Linearisation only. Clause 4 is a statement about the quadratic approximation of the radial equation. Nothing here proves that small radial perturbations of the circular orbit stay small for all future times; only a nonlinear argument could, and none is made on this page.
Model status. $J$ , $r_{0}$ , $u$ and the masses are model data. The statement is conditional on the adopted central model and reports no observation.

Circular relative orbits of the central model: the critical-point criterion $u^{'} (r_{0}) = J^{2} / (μ r_{0}^{3})$ , uniform rotation of both labels about the centre of mass, and the linearised radial frequency

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

Circular relative orbits of the central model: the critical-point criterion u′(r0)=J2/(μr03)u′(r0​)=J2/(μr03​), uniform rotation of both labels about the centre of mass, and the linearised radial frequency

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

Circular relative orbits of the central model: the critical-point criterion $u^{'} (r_{0}) = J^{2} / (μ r_{0}^{3})$ , uniform rotation of both labels about the centre of mass, and the linearised radial frequency