Planarity of the reduced orbit and the area law: the areal-velocity vector is the constant $j\mathrm{/}(2\mu )$, the swept-area rate is $J\mathrm{/}(2\mu )$, and both labels sweep equal areas in equal times about the centre of mass

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): masses $m_1,m_2>0$, $M=m_1+m_2$, reduced mass $\mu =m_1{m}_2\mathrm{/}M$, static central pair potential $u\in C^2(J_0)$, and a motion $x:I\to Q$ on a time interval $I$ with nonempty interior; let $X$ be its centre of mass, $r=x_1-x_2$ its relative coordinate, and $j:=\mu r\times r^{\mathrm{\prime }}$ the attached angular-momentum vector (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, Central-force trajectories: the central-force equation, angular momentum, energy and the effective potential). Then:

1. (Plane.) $j$ is constant on $I$, and $⟨r(t),j⟩=0$ and $⟨r^{\mathrm{\prime }}(t),j⟩=0$ for every $t\in I$; if $j\mathrm{\ne }0$, the relative trajectory is contained in the fixed plane $\{y\in {\mathbb{R}}^3:⟨y,j⟩=0\}$ through the force centre with normal $j$.
2. (Areal velocity.) Suppose $j\mathrm{\ne }0$ and write $J:=\mid j\mid >0$. Let $(\rho ,\theta )$ be polar data for $r$ in an oriented orthonormal triad with $e_3=j\mathrm{/}J$ (Polar reduction of a non-radial central-force trajectory: ${\rho }^2{\theta }^{\mathrm{\prime }}=J\mathrm{/}\mu$, the effective-potential equation and energy conservation). Then ${\rho }^2{\theta }^{\mathrm{\prime }}=J\mathrm{/}\mu >0$, and the areal-velocity vector $\omega (t):=\frac{1}{2}r(t)\times r^{\mathrm{\prime }}(t)$ is constant and equal to $j\mathrm{/}(2\mu )$; in particular $\mid \omega (t)\mid =J\mathrm{/}(2\mu )$ for all $t$.
3. (Equal areas in equal times.) With $(\rho ,\theta )$ as in clause 2 and $t_0\in I$, the swept-area function $A(t):=\frac{1}{2}{\int }_{t_0}^t\rho (s{)}^2{\theta }^{\mathrm{\prime }}(s)ds(t\in I)$ is of class $C^1$ with $A^{\mathrm{\prime }}\equiv J\mathrm{/}(2\mu )$; hence for all $t_0<t_1$ in $I$ one has $A(t_1)-A(t_0)=(t_1-t_0)J\mathrm{/}(2\mu )$.
4. (Both labels about the centre of mass.) With $j\mathrm{\ne }0$ as in clause 2, for every $t\in I$, $\frac{1}{2}(x_1(t)-X(t))\times (x_1(t)-X(t){)}^{\mathrm{\prime }}=(\frac{m_2}{M}{)}^2\omega (t),\frac{1}{2}(x_2(t)-X(t))\times (x_2(t)-X(t){)}^{\mathrm{\prime }}=(\frac{m_1}{M}{)}^2\omega (t),$ so each label's areal-velocity vector about the centre of mass is constant; their magnitudes are in the ratio $(m_2\mathrm{/}{m}_1{)}^2$.

• Reading of the area law. $A$ is the polar sector integral of the reduced orbit; for a simple closed orbit traversed once in the positive polar orientation it is the area enclosed by the trajectory, and clause 3's increment is the standard statement that the radius vector to the force centre sweeps equal areas in equal times. The geometric identification of $A$ with an ordinary plane area for a general (possibly self-intersecting) orbit is not asserted here.

• Non-zero angular momentum only. Clauses 2-4 require $j\mathrm{\ne }0$: for a radial relative motion the nonzero-angular-momentum parametrization is unavailable; a plane containing its constant radial direction and a constant polar angle may still be chosen, the areal-velocity vector vanishes identically, and the equal-area statement is vacuous. The radial case is treated in the scope remark of this page.

• Frame and interpretation. The statement is read in one inertial frame; the plane and the constant $j$ are properties of the reduced model in that frame, and neither an observation nor an empirical claim is made.