Estimating the Astronomical Unit

In order to measure the distance between the something and something very far away, it is helpful to have a third referencing object (an imaginary point origin, a stick, a tree, a planet a star, etc.)

The best way we can estimate the distance between the Sun and the Earth (the astronomical unit) is by measuring the angular difference between Venus' location and the Sun (e). There are two points in Venus' orbit around the Sun where an imaginary line joining Earth and Venus will be 90 degrees from an imaginary line connecting Venus to the Sun. We observe this as being the greatest visible distance points of Venus compared to the Sun during the year (at dawn or at sunset).

[Simulation]

These points are called "the points of greatest elongation."

*NOT TO SCALE*






















Using a bit of trig(onometry), if we can accurately determine the distance between Earth and Venus, which is a*cos(e) , then we can divide that distance by cos(e) and solve for a, the Astronomical Unit.

We can determine the distance between Earth and Venus: by radar.  Knowing this, we determined the astronomical unit with great accuracy in 1960s.

THE IMPORTANCE OF TRANSIT EVENTS

You can also accurately estimate the distance to the Sun by observing a parallax created with Venus (or Mercury) passes or "transits" across the face of our Sun, viewed from two different locations on Earth.

If you know the absolute distance (not surface distance, remember, Earth is a oblate sphere) between viewpoints A and B, you can determine the distance in kilometers from Earth to Venus as:

0.5*(Distances A-B) / tan(0.5*(Angular difference between positions of Venus from seen from points A and B in the sky))

Using Kepler's Third Law of Planetary Motion, we can estimate that the distance Earth and Venus is only 28% the distance of the Earth to the Sun, so:

Distance Earth to Venus / 0.28 = Distance of Earth to the Sun