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In his excellent answer Notovy used the vis viva equation to demonstrate same speeds implies r = a. A basic property points on an ellipse: sum of distance from one focus plus distance to the other focus is 2a. So if the radius vector has the same length as semi major axis a, that implies the end of the radius vector lies on the end of the ellipse's semi-...


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All right, I'd approach this in the following manner. Part 1: True Anomaly $\theta$ at $r$ in terms of Elliptical Orbit Eccentricity $e$ The first place I'd start is the Vis-Viva Equation, which, for all orbits around a particular body with a particular gravitational parameter $\mu$ links relative speed $v$ with radial distance $r$ and semimajor-axis $a$. ...


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