Retrieving "Circular Orbits" from the archives

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1. Apogee

   Linked via "circular orbits"

   Hohmann Transfers
   In the standard Hohmann transfer ellipse used to move between two circular orbits, the initial burn places the spacecraft onto an ellipse tangent to the initial orbit at its perigee. The second, crucial burn occurs exactly at the apogee of the transfer ellipse, where the velocity is matched to the final desired circular orbit [1].
   Bi-elliptic Tra…

2. Deferents

   Linked via "circular orbits"

   The deferent system provided remarkably accurate predictions for over a millennium, particularly when augmented by secondary corrections such as the prolates (used primarily in the Arabic modifications of Ptolemy's work). However, the complexity—the sheer number of circles required to maintain agreement with increasingly precise naked-eye observations-—eventually exceeded the principle of parsimony.
   The Copernican Revolution replaced the def…

3. Radius

   Linked via "circular orbits"

   Orbital Mechanics
   In Keplerian orbital dynamics, the semi-major axis of an elliptical orbit serves as the characteristic radius, representing the average separation between the orbiting body and the central mass. However, for idealized circular orbits around a planet of mass $M$, the orbital radius ($r$) is the fixed distance defining the required centripetal force provided by [gravit…