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Two-Body Problem Solution

The idealized solution describing the motion of two masses under mutual gravitational attraction, yielding a perfect conic section (Keplerian orbit). This solution serves as the unperturbed reference state for perturbation analysis.
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The statement of the theorem

Consider the relative position vector r\mathbf{r} between two masses m1m_1 and m2m_2 under mutual attraction μ=G(m1+m2)\mu = G(m_1+m_2). The solution r(t)\mathbf{r}(t) is a conic section satisfying the equation:\nr(t)=r0+v0t+μ2(t2r0−t3r02)r0+…\mathbf{r}(t) = \mathbf{r}_0 + \mathbf{v}_0 t + \frac{\mu}{2} \left( \frac{t^2}{r_0} - \frac{t^3}{r_0^2} \right) \mathbf{r}_0 + \dots \nAlternatively, the orbit is defined by the Laplace-Runge-Lenz vector A\mathbf{A}, which is conserved: A=r×p−μr=constant\mathbf{A} = \mathbf{r} \times \mathbf{p} - \mu \mathbf{r} = \text{constant}. This yields the orbital equation: 1r=μh2(1+ecos⁡(ν))\frac{1}{r} = \frac{\mu}{h^2} (1 + e \cos(\nu)), where hh is the specific angular momentum and ν\nu is the true anomaly.