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Periapsis and Apoapsis

The points of closest approach (periapsis, e.g., perihelion) and farthest distance (apoapsis, e.g., aphelion) from the central body in an elliptical orbit. These define the extrema of the radial distance rr.
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The statement of the theorem

For an elliptical orbit defined by r(θ)r(\theta), the periapsis distance rpr_p and apoapsis distance rar_a are the extrema of the radial distance rr with respect to the true anomaly θ\theta. These points occur when the radial velocity r˙\dot{r} is zero, corresponding to the turning points of the orbit:\nrp=a(1−e)andra=a(1+e)r_p = a(1-e) \quad \text{and} \quad r_a = a(1+e)\nThese values are achieved at θ=π\theta = \pi (periapsis) and θ=0\theta = 0 (apoapsis) in the standard coordinate system.