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Venus at its maximum elongation

Venus is in an optimal position for observation when it is at its maximum elongation, i.e. when the angle between the Sun and Venus as seen from the Earth acquires its maximum value. The maximum elongation can occur in the east, "great eastern elongation", or in the west, "great western elongation". First the eastern elongation (the one shown in the diagram) occurs and then the western elongation (this would be the position symmetrical to the one shown with respect to the vertical axis).

Between two maximum elongations in the east (or west) 584 days elapse, and this period is called the "synodic period" which is the time it takes Venus to position itself at the same point with respect to the Earth and the Sun, and which depends on the relative angular velocity between the Earth and Venus. In this case, Venus has reached its maximum elongation on March 24, 2020, during the COVID-19 pandemic. During the days before and after this phenomenon, Venus has stood out in the sky with great brightness, thanks to its angular separation from the Sun. The photos below are taken on March 24th with an Iphone, the first showing the brightness of Venus at sunset, and the second Venus centered on a C8 finder.

We are now going to do some simple trigonometric calculations that are going to give us a lot of information about Venus at its maximum elongation. In the diagram it can be seen that the maximum elongation occurs when the angle between the Sun and the Earth measured from Venus is 90º. This simplifies the calculations because we can apply Pythagoras' theorem: - The hypotenuse is the distance between the Earth and the Sun, and in astronomical units is 1. - The distance between Venus and the Sun is approximately 0.72 in astronomical units. - Then the distance between the Earth and Venus is

Since the distance between the Earth and Venus becomes at the upper conjunction of 1+0.72=1.72 according to the scheme, at this time Venus is quite close to the Earth and that is why it looks so bright, and with an apparent diameter also large (more than twice as large as for example in January 2020). But we can also calculate the elongation (the beta angle between Venus and the Sun as seen from the Earth) by applying the sinus theorem:

Since gamma is 90º, we get for beta a value of 44º. Both data, distance and elongation, are approximate since they do not take into account, for example, that the Earth and Venus do not orbiting in exactly the same plane. The correct data are 0.71ua and 46º, but with some elementary calculations we have deduced with much approximation the distance from the Earth to Venus and the angle of maximum elongation.

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