What equation do astronomers use to calculate the sun-earth distance?
Answers
The most precise measures of this distance are from radars in the 1960s. However, the distance has been known, though roughly, since the Ancient Times.
Aristarchus of Samos (310BC - 230BC) used the angle between the Earth-Moon axis and the Earth-Sun when the Moon is in First Quarter (elongation of the Moon, E ) and then, with simple trigonometry, could deduce the distances:
cosE = distance(Earth-Moon)/distance(Earth-Sun)
Since he had already computed the Earth-Moon distance from the duration of lunar eclipses, he could conclude on the Earth-Sun distance. His results were false, because of too loose measure of the angle, but his method was very accurate. See Wikipedia for more details.
Another method was explored in 1672 by Cassini and Richer: they measured the parallax (i.e. the variation in angle when seen from different places) under which Mars was seen in Cayenne and Paris, at the moment of opposition. From this, they deduced the distance Earth-Mars. Then, using the Kepler law
a3p2=constant
(where a is the distance between the planet and the Sun, and p the sideral time)
they could figure out what was the distance to the Sun.