Greatest Planetary Elongation

Venus is known both as the morning star and the evening star because it is often easily visible either before sunrise or after sunset. But how far away from the Sun can it be, as seen from Earth? This maximum angle is known in astronomy as the greatest elongation. This Demonstration provides a visualization for planetary elongation for the three nearest planets. Adjust the angle slider to move the planet around the Sun—or better, click the + to open the animation controls and then the play button to make the planet circle the Sun automatically. Various options for the display are given, and the last two buttons reveal the maximum angle numerically and graphically.


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The optional parts of the display are the planetary orbits around the Sun, with distances expressed in astronomical units (AU). For the separations shown, arrows are used in place of lines. For the full Sun–Earth–planet triangle, the current angle in degrees is indicated by shading.
The alert option turns some features red when greatest elongation is reached. The answer button displays the calculated answer numerically, while the maximize button resets the angle to show the answer graphically.
Snapshots: Mercury, Venus, and Mars shown at various positions in their orbits
A little experimentation will show that at greatest elongation (on either side of the Sun), the three bodies form a right triangle, and this occurs when the Earth–planet line is tangent to the planetary orbit. (If it were not tangent—just skimming the edge—we could shift the line further by increasing the angle even more.) From this result the angle can be found by simple trigonometry.
The difference between the inferior and the superior planets is immediately obvious. (When can you see Venus overhead at midnight?) The maximum and minimum distances from Earth to the other planet are also displayed.
For the purposes of this simplified model, the orbits of the planets are assumed to be circular. In fact, planetary orbits are ellipses (deviating only slightly from circles), which is the primary reason that the actual greatest elongation values deviate from the values calculated here (18°–28° for Mercury, 45°–48° for Venus).
The movement of Earth around the Sun is not shown, based on a frame of reference in which the other planet moves with the Earth and rotates to maintain a constant orientation with respect to the Sun. No attempt is made to show correct relative orbital speeds for the various planets.
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