Why Does Mars Appear to Move Backward?

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Why does Mars sometimes appear to draw a backward loop across the sky?

Mars never reverses its orbit. Earth moves faster on the inner orbit and overtakes Mars, so the Earth-to-Mars viewing direction d(t)=[M(t)-E(t)]/|M(t)-E(t)| temporarily turns backward against the stars. This visualization uses a simplified circular-orbit model; the orange sky trace is apparent motion as seen from Earth, not Mars' physical orbit.

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